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INTRODUCTION 


TO 


ASTRONOMY: 


DESIGNED  AS  A 


TEXT    BOOK 


FOR  THE 


STUDENTS  OF  YALE  COLLEGE. 


STEREOTYPE    EDITION. 


BY  DENISON  OLMSTED,  LL.  D., 

PROFESSOR  OF  NATURAL  PHILOSOPHY   AND  ASTRONOMY. 


NEW  YORK: 
ROBERT  B.   COLLINS. 
1850. 


Entered,  according  to  Act  of  Congress,  in  the  year  1839, 

By  DENISON  OLMSTED, 
In  the  Clerk's  Office  of  the  District  Court  of  Connecticut. 


Stereotyped  by 

RICHARD  C.  VALENTINE, 
45  Gold-street,  New  York. 


PREFACE. 


NEARLY  all  who  have  written  Treatises  on  Astronomy,  designed  for  young 
learners,  appear  to  have  erred  in  one  of  two  ways ;  they  have  either  disre- 
garded demonstrative  evidence,  and  relied  on  mere  popular  illustration,  or  they 
have  exhibited  the  elements  of  the  science  in  naked  mathematical  formula?. 
The  former  are  usually  diffuse  and  superficial ;  the  latter,  technical  and  ab- 
struse. 

In  the  following  Treatise,  we  have  endeavored  to  unite  the  advantages  of 
both  methods.  We  have  sought,  first,  to  establish  the  great  principles  of 
astronomy  on  a  mathematical  basis  ;  and,  secondly,  to  render  the  study  inter- 
esting and  intelligible  to  the  learner,  by  easy  and  familiar  illustrations.  We 
would  not  encourage  any  one  to  believe  that  he  can  enjoy  a  full  view  of  the 
grand  edifice  of  astronomy,  while  its  noble  foundations  are  hidden  from  his 
sight;  nor  would  we  assure  him  that  he  can  contemplate  the  structure  in  its 
true  magnificence,  while  its  basement  alone  is  within  his  field  of  vision.  We 
would,  therefore,  that  the  student  of  astronomy  should  confine  his  attention 
neither  to  the  exterior  of  the  building,  nor  to  the  mere  analytic  investigation 
of  its  structure.  We  would  desire  that  he  should  not  only  study  it  in  models 
and  diagrams,  and  mathematical  formulas,  but  should  at  the  same  time  acquire 
a  love  of  nature  herself,  and  cultivate  the  habit  of  raising  his  views  to  the 
grand  originals.  Nor  is  the  effort  to  form  a  clear  conception  of  the  motions  and 
dimensions  of  the  heavenly  bodies,  less  favorable  to  the  improvement  of  the 
intellectual  powers,  than  the  study  of  pure  geometry. 

But  it  is  evidently  possible  to  follow  out  all  the  intricacies  of  an  analytical 
process,  and  to  arrive  at  a  full  conviction  of  the  great  truths  of  astronomy,  and 
yet  know  very  little  of  nature.  According  to  our  experience,  however,  but  few 
students  in  the  course  of  a  liberal  education  will  feel  satisfied  with  this.  They 
do  not  need  so  much  to  be  convinced  that  the  assertions  of  astronomers  are 
true,  as  they  desire  to  know  what  the  truths  are,  and  how  they  were  ascer- 
tained ;  and  they  will  derive  from  the  study  of  astronomy  little  of  that  moral 
and  intellectual  elevation  which  they  had  anticipated,  unless  they  learn  to  look 
upon  the  heavens  with  new  views,  and  a  clear  comprehension  of  their  won- 
derful mechanism. 

Much  of  the  difficulty  that  usually  attends  the  early  progress  of  the  astro- 
nomical student,  arises  from  his  being  too  soon  introduced  to  the  most  perplex- 
ing part  of  the  whole  subject, — the  planetary  motions.  In  this  work,  the  con- 
sideration of  these  is  for  the  most  part  postponed  until  the  learner  has  become 
familiar  with  the  artificial  circles  of  the  sphere,  and  conversant  with  the  celes- 
tial bodies.  We  then  first  take  the  most  simple  view  possible  of  the  planetary 
motions  by  contemplating  them  as  they  really  are  in  nature,  and  afterwards 
proceed  to  the  more  difficult  inquiry,  why  they  appear  as  they  do.  Probably 
no  science  derives  such  signal  advantage  from  a  happy  arrangement,  as  as- 
tronomy ; — an  order,  which  brings  out  every  fact  or  doctrine  of  the  science  just 
in  the  p'lace  where  the  mind  of  the  learner  is  prepared  to  receive  it. 

Although  we  have  found  it  convenient  to  defer  the  consideration  of  the  fixed 
stars  to  a  late  period,  yet  we  would  earnestly  recommend  to  the  student  to  be- 
gin to  learn  the  constellations,  and  the  stars  of  the  first  magnitude  at  least,  as 


IV  PREFACE. 

soon  as  he  enters  upon  the  study  of  astronomy.  A  few  evenings  spent  in  this 
way,  assisted,  where  it  is  practicable,  by  a  friend  already  conversant  with  the 
stars,  will  inspire  a  higher  degree  of  enthusiasm  for  the  science,  and  render  its 
explanations  more  easily  understood. 

It  is  recommended  to  the  learner  to  make  a  free  use  of  the  Analysis,  espe- 
cially in  reviewing  the  ground  already  traversed.  If  by  repeated  recurrence  to 
these  heads,  he  associates  with  each  a  train  of  ideas,  carrying  along  with  him, 
as  he  advances,  all  the  particulars  indicated  in  these  hints,  he  will  secure  to 
them  an  indelible  place  in  his  memory. 

With  such  aids  at  hand,  as  Newton,  La  Place,  and  Delambre,  to  expound 
the  laws  of  astronomy,  and  such  popular  writers  as  Ferguson,  Biot,  and  Fran- 
cceur,  to  supply  familiar  illustrations  of  those  laws,  it  might  seem  an  easy  task 
to  prepare  a  work  like  the  present ;  but  a  text  book  made  up  of  extracts  from 
these  authors,  would  be  ill  suited  to  the  wants  of  our  students.  We  have 
deemed  it  better  therefore,  first,  to  acquire  the  clearest  views  we  were  able  of 
the  truths  to  be  unfolded,  both  from  an  extensive  perusal  of  standard  authors, 
and  from  diligent  reflection,  and  then  to  endeavor  to  transfuse  our  own  im- 
pressions into  the  mind  of  the  learner.  Writers  of  profound  attainments  in 
astronomy,  and  of  the  highest  reputation,  have  often  failed  in  the  preparation 
of  elementary  works,  because  they  lacked  one  qualification — the  experience  of 
the  teacher.  Familiar  as  they  were  with  the  truths  of  the  science,  but  unac- 
customed to  hold  communion  with  young  pupils,  they  were  incapable  of  ap- 
prehending the  difficulty  and  the  slowness  with  which  these  truths  make  their 
entrance  into  the  mind  for  the  first  time.  Even  when  they  attempt  to  feel 
their  way  into  young  minds,  by  assuming  the  garb  of  the  instructor,  and  em- 
ploying popular  illustrations,  they  often  betray  their  want  of  the  experience 
and  art  of  the  professional  teacher. 

Astronomy,  in  its  grandest  and  noblest  conceptions,  addresses  itself  alike  to 
the  intellect  and  to  the  heart.  It  demands  the  highest  efforts  of  the  one  and  the 
warmest  and  most  devout  affections  of  the  other,  in  order  fully  to  comprehend 
its  truths  and  to  relish  its  sublimity.  The  task  of  learning  the  bare  elements 
of  this,  as  well  as  of  every  other  science,  is  purely  intellectual,  and  is  to  be  re- 
garded only  as  preparing  the  way  for  that  more  enlarged  and  exalted  contem- 
plation of  the  heavenly  bodies,  to  which  the  mind  will  naturally  rise,  when  it 
can  view  all  things  in  their  true  relations  to  each  other.  It  is  therefore  essen- 
tial to  this  study,  as  a  part  of  a  public  education,  that  the  student,  after  ac- 
quiring a  knowledge  of  the  elements  of  the  science,  should  return  to  the  sub- 
ject, and  trace  the  great  discoveries  of  astronomy,  as  they  have  succeeded  one 
another  from  the  earliest  ages  of  society  down  to  the  present  time,  viewing  them 
in  connection  with  the  many  interesting  historical  and  biographical  incidents 
which  attended  their  development.  The  author  is  therefore  accustomed,  in 
his  own  course  of  instruction,  to  follow  the  study  of  this  "Introduction,"  with 
a  course  of  Lectures  adapted  to  such  a  purpose  ;  and,  with  similar  views,  he 
has  prepared  a  volume  of "  Letters  on  Astronomy,"  where  he  has  attempted 
to  connect  with  the  leading  truths  of  the  science  such  historical  incidents  and 
moral  reflections,  as  may  at  once  interest  the  understanding  and  amend  the 
heart. 


ANALYSIS. 


DESIGNED  AS  A  BASIS  FOR  REVIEW  AND  EXAMINATION. 


PRELIMINARY   OBSERVATIONS. 

Astronomy  defined, 

Descriptive  Astronomy, 

Physical  do.  

Practical  do.  

History. — Ancient  nations  who  cultiva- 
ted astronomy, 

Pythagoras — his  age  and  country, 1 

His  views  of  the  celestial  motions, 

Alexandrian  School — when  founded — 
by  whom — introduction  of  astronomi- 
cal instruments, 2 

Hipparchus — his  character, 2 

Ptolemy — the  Almagest, 2 

Copernicus,  Tycho  Brahe,  Kepler  and 
Galileo — respective  labors  of  each,....  2 

Sir  Isaac  Newton — his  great  discovery,     2 

La  Place — Mecanique  Celeste, 2 

Astrology — Natural  and  Judicial — ob- 
ject of  each, 2 

Accuracy  aimed  at  by  astronomers, 

Copernican  System — its  leading  doc- 
trines,   3 

Plan  of  this  work, 3 


Part  I.— OF  THE  EARTH. 

Chapter  1. — OF  THE  FIGURE  AND  DIMENSIONS 

OF  THE  EARTH,  AND  THE  DOCTRINE 
SPHERE. 

Figure  of  the  earth, 4 

Proofs, 4 

Dip  of  the  horizon, 4 

How  found, 5 

Table  of  the  dip — its  use, 6 

Exact  figure  of  the  earth, 6 

Its  circumference, 6 

Small  inequalities  of  the  earth's  surface,  6 

Diameter  of  the  earth  how  determined,  7 
How  to  divest  the  mind  of  preconceived 

erroneous  notions, 8 

DOCTRINE  OF  THE  SPHERE,  defined, 9 

Great  and  small  circles  defined, 9 

Axis  of  a  circle — pole, 9 

Situation  of  the  poles  of  two  great  cir- 
cles which  cut  each  other  at  right  an- 

gles, 9 

Points  of  intersection  of  two  great  cir- 
cles—how many  degrees  apart, 


Page. 

Page.  When  a  great  circle  passes  through  the 

1      pole  of  another,  how  does  it  cut  it  ? .  10 

1  Secondary  defined, 10"""" 

1  Angle  made  by  two  great  circles  how 

1      measured, 10 

Terrestrial  and  Celestial  spheres  distin- 

1      guished, 10 

Horizon  defined, 11 

Sensible  horizon, 11 

Rational     do 11 

Zenith  and  Nadir, 11 

Vertical  circles, 11 

Meridian, 11 

Prime  Vertical, 11 

How  the  place  of  a  celestial  body  is  de- 
termined,   11 

Altitude — azimuth — amplitude, 12 

Zenith  Distance — how  measured, 12 

Axis  of  the  earth — axis  of  the  celestial 

sphere, 12 

Poles  of  the  earth — poles  of  the  heav- 
ens,   12 

Equator — terrestrial  and  celestial, 12 

Hour  circles, 

Latitude, 13 

Polar  Distance,  how  related  to  latitude,  13 

Longitude, 13 

OF  THE  Standard  Meridians, 13 

Ecliptic, 13 

Inclination  of  the  ecliptic  to  the  equa- 
tor,   13 

Equinoctial  points, 13 

Equinoxes — Vernal  and  autumnal, ....  13 

Solstitial  points, 14 

Solstices, 14 

Signs  of  the  ecliptic  enumerated, 14 

3olures — Equinoctial  and  Solstitial,...  14 

Right  ascension, 15 

Declination, 15 

elestial  Longitude, 15 

Celestial  Latitude, 15 

North  Polar  Distances,  how  related  to 

latitude, 15 

Parallels  of  Latitude, 15 

Tropics, 16 

Polar  circles, , 16 

16 

16 


Ziones, 

Zr 


10  Zodiac,.. 


ANALYSIS. 


Page. 
Elevation  of  the  pole — to  what  is  it 

equal? 16 

Elevation  of  the  equator, 16 

Distance  of  a  place  from  the  pole,  to 

what  equal? 16 

Chapter   II. — DIURNAL   REVOLUTION — ARTI- 
FICIAL GLOBES ASTRONOMICAL  PROBLEMS. 

Circles  of  Diurnal  Revolution, 17 

Sidereal  day  defined, 17 

Appearance  of  the  circles  of  diurnal 

revolution  at  the  equator, 17 

ARight  Sphere  defined, 18 

A  Parallel  Sphere, 19 

An  Oblique  Sphere, 19 

Circle  of  Perpetual  Apparition, 20 

Circle  of  Perpetual  Occultation, 20 

How  are  the  circles  of  daily  motion  cut 
by  the  horizon  in  the  different 

spheres? 20 

Explanation  of  the  peculiar  appearan- 
ces of  each  sphere,  from  the  revolu- 
tion of  the  earth  on  its  axis, 21 

tificial  Globes — terrestrial  and  celes- 
tial,      22 

Their  use, 23 

Meridian — how  represented — how  gra- 
duated,   23 

Horizon — how  represented — how  gra- 
duated,   23 

Hour  Circles,  how  represented, 23 

Hour  Index  described, 23 

Quadrant  of  Altitude, 24 

Its  use  described, 24 

To  rectify  the  globe  for  any  place, 24 

PROBLEMS  ON  THE  TERRESTRIAL  GLOBE 
— To  find  the  latitude  and  longitude 

of  a  place, 24 

To  find  a  place,  its  latitude  and  longi- 

gitude  being  given, 25 

To  find  the  bearing  and  distance  of 

two  places, 25 

To  determine  the  difference  of  time  of 

two  places, 25 

The  hour  being  given  at  any  place,  to 
tell  what  hour  it  is  in  any  other  part 

oftheworld, 25 

To  find  the  antceci,  periaeci,  and  antipo- 
des,   25 

To  rectify  the  globe  for  the  sun's  place,    26 
The  latitude  of  the  place  being  given, 
to  find  the  time  of  the  sun's  rising 

and  setting, 26 

PROBLEMS  ON  THE  CELESTIAL  GLOBE. — 
To  find  the  right  ascension  and  decli- 
nation,  •. 26 

To  represent   the    appearance  of  the 

heavens  at  any  time, 2' 

To  find  the  altitude  and  azimuth  of  a 
star, 2 


Page. 

To  find  the  angular  distance  of  two 
stars  from  each  other, 27 

To  find  the  surfs  meridian  altitude, 
the  latitude  and  day  of  the  month 
being  given, 28 

Chapter  III. — PARALLAX — REFRACTION — 
TWILIGHT. 

*arallax  defined, 28 

lorizontal  Parallax, 29 

lelation  of  parallax  to  the  zenith  dis- 
tance, and  distance  from  the  center 

of  the  earth, 29 

To  find  the  horizontal  parallax  from 

the  parallax  at  any  altitude, 29 

Amount  of  parallax  in  the  zenith  and 

in  the  horizon, 30 

fFect  of  parallax  upon  the  altitude  of 

a  body, , 30 

Mode  of  determining   the   horizontal 

parallax  of  a  body, 30 

Amount  of  the  sun's  hor.  par 31 

Jse  of  parallax, 31 

Refraction. — Its  effect  upon  the  alti- 
tude of  a  body, 32 

[ts  nature  illustrated, 32 

[ts  amount  at  different  angles  of  eleva- 
tion,   32 

How  the  amount  is  ascertained, 33 

Sources  of  inaccuracy  in  estimating  the 

refraction, 35 

Effect  of  refraction  upon  the  sun  and 

moon  when  near  the  horizon, 35 

Oval  figure  of  these  bodies  explained,.     35 
Apparent  enlargement  of  the  sun  and 

moon  near  the  horizon, 36 

Twilight. — Its  cause  explained, 37 

Length  of  twilight  in  different  latitudes,     37 
How  the  atmosphere  contributes  to  dif- 
fuse the  sun's  light, 37 

Chapter  IV.— TIME. 

Time  defined, 38 

What  period  is  a  sidereal  day, 38 

Uniformity  of  sidereal  days, 38 

Solar  time,  how  reckoned, 

Why  solar  days  are  longer  than  side- 


real, 


39 

Apparent  time  defined, 39 

Mean  time, 40 

An  astronomical  day, 40 

Equation  of  time  defined, 40 

When  do  apparent  time  and  mean 

time  differ  most? 40 

When  do  they  come  together? 40 

Effect  of  a  change  in  the  place  of  the 

earth's  perihelion, 40 

Causes  of  the  inequality  of  the  solar 

days, 41 

Explain  the  first  cause,  depending  on 

the  unequal  velocities  of  the  sun,....  41 


ANALYSIS. 


Vii 


Explain  the  second  cause,  depending 

on  the  obliquity  of  the  ecliptic, 42 

When    does    the   sidereal    day    com- 

mence? 44 

The  Calendar. — Astronomical  year  de- 
fined,   45 

How  the  most  ancient  nations  deter- 

mined  the  number  of  days  in  the  year,  45 
Julius  Caesar's  reformation  of  the  calen. 

dar  explained, 45 

Errors  of  this  calendar, 45 

Reformation  by  Pope  Gregory 46 

Rule  for  the  Gregorian  calendar, 46 

New  style,  when  adopted  in  England,  46 
What  nations  still  adhere  to  the  old 

style? 46 

What  number  of  days  is  now  allowed 

between  old  and  new  style  ? 47 

How  the  common  year  begins  and  ends,  47 

How  leap  year  begins  and  ends, 47 

Does  the  confusion  of  different  calen- 
dars affect  astronomical  observations  ?  47 

Chapter  V. — ASTRONOMICAL   INSTRUMENTS 
AND  PROBLEMS — FIGURE  AND  DENSITY  OF 


THE  EARTH. 

How   the  most   ancient  nations   acquired 

their  knowledge  of  Astronomy, 48 

Use  of  instruments  in  the  Alexandrian 

School, 48 

Ditto,  by  Tycho  Brahe, 48 

Ditto,  by  the  Astronomers  Royal, 48 

Space  occupied  by  l"on  the  limb  of  an 

instrument, 48 

Extent  of  actual  divisions  on  the  limb,  49 

Vernier,  defined, 49 

Its  use  illustrated, 49 

Chief  astronomical   instruments  enu- 
merated,   50 

Observations  taken  on  the  meridian. . .  50 

Reasons  of  this, 50 

Transit  Instrument  defined, 51 

Ditto                  described, 51 

Method  of  placing  it  in  the  meridian. .  51 

Line  of  collimation  defined, 52 

System  of  wires  in  the  focus, 52 

Its  use  for  arcs  of  right  ascension, 52 

Astronomical   Clock, — how  regulated,  52 

What  does  it  show  ? 

How  to  test  its  accuracy, 53 

How  corrected, 

Mural  Circle,  its  object, 54 

Describe  it, 54 

How  the  different  parts  contribute  to 

theobject,. 54 

Mural  Quadrant, 

Use  of  the  Mural  Circle  for  arcs  of  de- 
clination,   56 

Altitude  and  Azimuth  Instrument  de- 
fined,...                                              ,  56 


Page.  Page. 

Its  use, 56 

Describe  it, 57 

Sextant  described, 58 

How  to  measure  the  angular  distance 

of  the  moon  from  the  sun, 59 

How  to  take  the  altitude  of  a  heavenly 

body -  59 

Use  of  the  arti  ficial  horizon , 59 

In  what  consists  the  peculiar  value  of 

the  Sextant? 60 

ASTRONOMICAL  PROBLEMS. — Given  the 
the  sun's  right  ascension  and  decli- 
nation, to  find  his  longitude  and  the 

obliquity  of  the  ecliptic, 61 

Napier's  Rule  of  circular  parts, 62 

Given  the  sun's  declination  to  find  his 
rising  and  setting  at  any  place  whose 

latitude  is  known, 63 

Given  the  latitude  of  a  place  and  the 
declination  of  a  heavenly  body,  to 
determine  its  altitude  and  azimuth 
when  on  the  six  o'clock  hour  circle,  64 
The  latitudes  and  longitudes  of  two 
celestial  objects  being  given,  to  find 

their  distance  apart, 65 

FIGURE  AND  DENSITY  OF  THE  EARTH — 
reason  for  ascertaining  it  with  great 

precision, 66 

How  found  from  the  centrifugal  force,  66 
From  measuring  an  arc  of  the  meridian,  67 
From  observations  with  the  pendulum,  68 

From  the  motions  of  the  moon, 68 

Density  of  the  earth  compared  with 

water, 68 

How  ascertained  by  Dr.  Maskelyne, . .  69 
Why  an  important  element, 69 

Part  II.— OF  THE  SOLAR   SYSTEM 

Chapter  1.— THE  SUN— SOLAR  SPOTS— Zo 
DIACAL  LIGHT. 

Figure  of  the  sun, 70 

Angle  subtended  by  a  line  of  400  miles,  70 

Distance  from  the  earth, 70 

Illustrated  by  motion  on  a  railway  car,  70 
Apparent  diameter  of  the   sun — new- 
found,    72 

How  to  find  the  linear  diameter, 71 

ow  much  larger  is  the  sun  than  the 

earth, 71 

53  Its  density  and  mass  compared  with 

the  earth's, 71 

Weight  at  the  surface  of  the  sun, 72 


52  H 


Velocity  of  falling  bodies  at  the  sun ...  72 

SOLAR  SPOTS. — Their  number, 72 

55  Size, 72 

Description, 72 

What  region  of  the  sun  do  they  oc- 
cupy,   73 

Proof  that  they  are  on  the  sun, 73 


viii 


ANALYSIS. 


Page 


How  we  learn  the  revolution  of  the  sun 

on  his  axis, 73 

Time  of  the  revolution, 73 

Apparent  paths  of  the  spots, — 

Inclination  of  the  solar  axis, 74 

Sun's  Nodes — when  does  the  sun  pass 

them  ? 75 

Cause  of  the  solar  spots, 76 

Faculae, 76 

ZODIACAL  LIGHT. — Where  seen, 76 

Its  form, 

Aspects  at  different  seasons, 

Its  motions, 

Its  nature, 


Product  of  the  angle  described  in  any 
given  time  by  the  square  of  the  dis- 
tance,   88 

74  Space  described  by  the  radius  vector  of 

the  solar  orbit  in  equal  times, 88 

How  to  represent  the  sun's  orbit  by  a 
diagram, 89 

Chapter  III. — UNIVERSAL  GRAVITATION. 


77  H 


Chapter  II. — APPARENT  ANNUAL 

OF  THE   SUN — SEASONS — FIGURE  OF  THE 
EARTH'S  ORBIT. 


76  Universal  Gravitation  defined, 90 

76  Why  is  it  called  attraction, 90 

istory  of  its  discovery, 90 

ow  was  the  gravitation  of  the  moon 

to  the  earth  first  inferred  ? 91 

MOTION  Laws  of  Gravitation. — If  a  body  re- 
volves about  an  immovable  center 


77  H 


78  If 


791 


Apparent  motion  of  the  sun , 

How  both  the  sun  and  earth  are  said  to 
move  from  west  to  east, 79 

Nature  and  position  of  the  sun's  orbit, 
how  determined, 

Changes  in  declination  how  found, 79 

Ditto,  in  right  ascension, 

Inferences  from  a  table  of  the  sun's  de- 
clinations,   80 

Ditto,  of  right  ascensions, 81 

Path  of  the  sun,  how  proved  to  be  a 
great  circle, 81 

Obliquity  of  the  ecliptic,  how  found,     81 

How  it  varies, 81 

Great  dimensions  of  the  earth's  orbit,    81 

Earth's  daily  motion  in  miles, 

Ditto,  hourly  ditto, 

Diurnal  motion  at  the  equator  per 
hour, 

SEASONS. — Causes  of  the  change  of  sea- 
sons,  

How  each  cause  operates, 

Illustrated  by  a  diagram, 

Change  of  seasons  had  the  equator  been 
perpendicular  to  the  ecliptic, 84 

FIGURE  OF  THE  EARTH'S  ORBIT. — Proof 
that  the  earth's  orbit  is  not  circular,  85 

Radius  vector  defined, 

Figure  of  the  earth's  orbit  how  ob- 
tained,  

Relative  distances  of  the  earth  from  the 
sun,  how  found, 86 

Perihelion  and  Aphelion  defined, 87 

Variations  in  the  sun's  apparent  diame- 
ter,... .  87 


Angular  velocities  of  the  sun  at  the  pe- 
rihelion and  aphelion, 

Ratio  of  these  velocities  to  the  dis- 


tances,  

How  to  calculate  the  relative  distances 
of  the  earth  from  the  sun's  daily  mo- 
tions,   88 


Page 


of  force,  and  is  constantly  attracted 

to  it,  how  will  itmove? 92 

a  body  describes  a  curve  around  a 
center  towards  which  it  tends  by  any 
force,  how  is  its  angular  velocity  re- 
lated to  the  distance, 93 

n  the  same  curve,  the  velocity  at  any 
point  of  the  curve  varies  as  what  ?     93 

80  If  equal  areas  be  described  about  a  cen- 
ter in  equal  times,  to  what  must  the 

force  tend? 94 

How  is  the  distance  of  any  planet  from 
the  sun  at  any  point  in  its  orbit,  to 
its  distance  from  the  superior  focus  ?  94 
Dase  of  two  bodies  gravitating  to  the 
same  center  where  one  descends  in  a 
straight  line,  and  the  other  revolves 
in  a  curve, 95 

82  Velocity  of  a  body  at  any  point  when 

falling  directly  to  the  sun, 97 

82  Relation  between  the  distances  and  pe- 
riodic times, 99 

82  Kepler's  three  great  laws, 99 

82  MOTION  IN  AN  ELLIPTICAL  ORBIT, 100 

83  Idea  of  a  projectile  force, 100 

Mature  of  the  impulse  originally  given 

to  the  earth, 100 

Two  forces  under  which  a  body  re- 
volves,   100 

85  Illustrated  by  the  motion  of  a  cannon 

ball, 101 

86  Why  a  planet  returns  to  the  sun, 102 

Illustration  by  a  suspended  ball, 103 

Chapter  IV. — PRECESSION  OF  THE  EQUI- 
NOXES— NUTATION — ABERRATION — MEAN 
AND  TRUE  PLACES  OF  THE  SUN. 


87  Precession  of  the  Equinoxes  defined,  104 
Why  so  called, 104 

87  Amount  of  Precession  annually, 104 

Revolution  of  the  equinoxes, 104 

Revolution  of  the  pole  of  the  equator 
around  the  pole  of  the  ecliptic, 105 


ANALYSIS. 


Page 
Changes  among  the  stars  caused  by 

precession, 105 

The  present  pole  star  not  always  such,     105 
What  will  be  the  pole  star  13,000 

years  hence  ? 105 

Cause  of  the  precession  of  the  equi- 
noxes,... 105 


Explain  how  the  cause  operates, 

Proportionate  effect  of  the  sun  and 
moon  in  producing  precession, 


106 

107 

Tropical  year  defined, 107 

How  much  shorter  than  the  sidereal 

year, 107 

Use  of  the  precession  of  the  equinoxes 

in  chronology, 107 

NUTATION,  defined, 108 

Explain  its  operation, 108 

Cause  of  Nutation, 108 

ABERRATION,  defined, 108 


109 
109 
109 

109 

Direction  of  this  motion, 110 


Illustrated  by  a  diagram, . 

Amount  of  aberration, 

Effect  on  the  places  of  the  stars, 

MOTION  OF  THE  APSIDES,  the  fact  sta- 
ted,, 


Time  of  revolution  of  the  line  of  Ap- 
sides,    110 

Present  longitude  of  the  perihelion,. .  110 

When  was  it  nothing  ? 110 

MEAN  AND  TRUE  PLACES  OF  THE  SUN,  111 

Mean  Motion  defined, Ill 

Illustrated  by  surveying  a  field, Ill 

Mean  and  true  longitude  distinguish- 


ed, 


111 


Equations  defined,  ........................  Ill 

Their  object,  ...............................  Ill 

Mean  and  True  anomaly  defined,  ....  112 

Equation  of  the  Center,  .................  112 

Explain  from  the  figure,  ................  112 

Chapter  V.  —  THE  MOON  —  LUNAR  GEOGRA- 
PHY —  PHASES  OF  THE  MOON  —  HER  REVO- 


LUTIONS. 


113 


Distance  of  the  moon  from  the  earth, 

Her  mean  horizontal  parallax,  .........  113 

Her  diameter,  ..............................  113 

Volume,  density,  and  mass,  ............  113 

Shines  by  reflected  light,  ................  113 

Appearance  in  the  telescope,  ...........  113 

Terminator  defined,  ......................  113 

Its  appearanpe,  ............................  113 

Proofs  of  Valleys,  .........................  114 

Form  of  these  .............................  114 

Best   time   for  observing   the   lunar 

mountains  and  valleys,  ...............  114 

Names  of  places  on  the  moon  double,  115 

Dusky  regions  how  named,  .............  115 

Point  out  remarkable  places  on  the 

map  of  the  moon,  ......................  115 

Explain  the  method  of  estimating  the 


Page. 
Specify    the    heights  of    particular 

mountains, 117 

Volcanoes,  proof  of  their  existence,... .     117 

Has  the  moon  an  atmosphere  ? 117 

Improbability  of  identifying  artificial 

structures  in  the  moon, 117 

PHASES  OF  THE  MOON,  their  cause,....     118 
Successive  appearances  of  the  moon 
from  one  new  moon  to  another, ....     118 

Syzygies  defined, 118 

Explain  the  phases  of  the  moon  from 

figure  46, 119 

REVOLUTIONS  OF  THE  MOON.     Period 
of  her  revolutions  about  the  earth, .     119 

Her  apparent  orbit  a  great  circle, 120 

A  sidereal  month  defined, 120 

A  synodical        do.  120 

Length  of  each, 120 

Why  the  synodical  is  longer, 120 

How  eachis  obtained, 120 

Inclination  of  the  lunar  orbit, 121 

Nodes  defined, 121 

Why  the  moon  sometimes  runs  high 

and  sometimes  low, 121 

Harvest  moon  defined, 122 

Ditto  explained, 122 

Explain  why  the  moon  is  nearer  to  us 
when  on  the  meridian  than  when 

near  the  horizon, 122 

Time  of  the  moon's  revolution  on  its 

axis, 123 

How  known, 123 

Librations  explained, 123 

Diurnal  Libration, 124 

Length  of  the  Lunar  days, 124 

Earth  never  seen  on  the  opposite  side 

of  the  moon, 124 

Appearances  of  the  earth  to  a  specta- 
tor on  the  moon, 124 

Why  the  earth  would  appear  to  re- 

main  fixed, 125 

Ascending  and  descending  nodes  dis- 
tinguished,       125 

Whether  the  earth  carries  the  moon 

around  the  sun, 126 

How  much  more  is  the  moon  attract- 
ed towards  the  sun  than  towards 

the  earth, 126 

When  does  the  sun  act  as  a  disturbing 

force  upon  the  moon  ? 126 

Why  does  not  the  moon  abandon  the 

earth  at  the  conjunction  ? 126 

The  moon's  orbit  concave  towards  the 

sun, 127 

How  the  elliptical  motion  of  the  moon 
about  the  earth  is  to  be  conceived 

of, 127 

Illustrations, 127 

Chapter  VI.— LUNAR  IRREGULARITIES. 


height  of  lunar  mountains, 115  Specify  their  general  cause, 127 


ANALYSIS. 


Unequal  action  of  the  sun  upon  the 
earth  and  moon, 

Oblique  action  of  earth  and  sun, 

Gravity  of  the  moon  towards  the 
earth  at  the  syzygies, 

Gravity  at  the  quadratures, 

Explain  the  disturbances  in  the 
moon's  motions  from  figure  48, 

Figure  of  the  moon's  orbit, 

How  its  figure  is  ascertained, 

Moon's  greatest  and  least  apparent  di- 
ameters,  

Her  greatest  and  least  distances  from 
the  earth, 

Perigee  and  Apogee  defined, 

Eccentricities  of  the  solar  and  lunar 
orbits  compared, 

Moon's  nodes,  their  change  of  place,. 

Rate  of  this  change  per  annum, , 

Period  of  their  revolution, 

Irregular  curve  described  by  the 
moon, 

Cause  of  the  retrograde  motion  of 
nodes, 

Explain  from  figure  50, 

Synodical  revolution  of  the  node  de- 
fined,  

Its  period, 

The  Saros  explained, 

The  Metonic  Cycle, 

Golden  Number, 

Revolution  of  the  line  of  apsides, 

Its  period, 

How  the  places  of  the  perigee  may  be 
found, 

Moon's  anomaly  defined, 

Cause  of  the  revolution  of  the  apsides, 

Amount  of  the  equation  of  the  Center, 

Evection  defined, 

Its  cause  explained, 

Variation  defined, 

Its  cause, 

Annual  Equation  explained, 

How  these  irregularities  were  first 
discovered, 

How  many  equations  are  applied  to 
the  moon's  motions  ? 

Method  of  proceeding  in  finding  the 
moon's  place, 

Successive  degrees  of  accuracy  at- 
tained,  

Periodic  and  secular  irregularities  dis- 
tinguished,  

Acceleration  of  the  moon's  mean  mo- 
tion explained, 

Its  consequences, 

Lunar  inequalities  of  latitude  and 

parallax, 

Chapter  VII. — ECLIPSES. 

Eclipse  of  the  moon,  when  it  happens, 


128 
128 

129 
129 

130 
132 
132 

13! 
132 
13: 
132 

133 
133 
133 
133 

133 

133 
134 

135 
135 
135 
135 
136 
136 
136 

136 
136 
136 
137 
137 
138 
140 
140 
140 

141 
141 
141 
141 
141 

141 
142 

142 
143 


Eclipse  of  the  sun,  when  it  happens, .     143 

When  only  can  each  occur, 143 

Why   an  eclipse  does  not  occur  at 

every  new  and  full  moon, 144 

Why  eclipses  happen  at  two  opposite 

months, 144 

Circumstances  which  affect  the  length 

of  the  earth's  shadow, 144 

Semi-angle  of  the  cone  of  the  earth's 

shadow,  to  what  equal, 145 

Length  of  the  earth's  shadow, 145 

Its   breadth     where   it    eclipses    the 

moon, 146 

Lunar  ecliptic  limit  defined, 146 

Solar,  ditto  146 

Amount  of  thelunar  ecliptic  limit,....     146 

Appulse  defined, 147 

Partial,   total,   central,  eclipse,  each 

defined, 147 

Penumbra  defined,. 147 

Semi-angle  of  the  moon's  penumbra, 

to  what  equal, 148 

Semi-angle  of  a  section  of  the  penum- 
bra where  the  moon  crosses  it, 148 

Moon's  horizontal  parallax  increased 

,why, 148 

Why  the  moon  is  visible  in  a  total 

eclipse, 148 

alculation  of  eclipses,  general  mode 
of  proceeding, 149 

To  find  the  exact  time  of  the  begin- 
ning, end,  duration,  and  magnitude 
of  a  lunar  eclipse,  by  figures  53,  54,  150 

Elements  of  an  eclipse  defined, 151 

Digits  defined, 153 

How  the  shadow  of  the  moon  travels 
over  the  earth  in  a  solar  eclipse,;...  153 

Why  the  calculation  of  a  solar  eclipse 

is  more  complicated  than  a  lunar,.     154 

Velocity  of  the  moon's  shadow, 154 

Different  ways  in  which  the  shadow 
traverses  the  earth,  according  as 
the  conjunction  is  near  the  node  or 
near  the  limit, 155 

When  do  the  greatest  eclipses  hap- 
pen ? 155 

ase   in    which  the  moon's  shadow 
nearly  reaches  the  earth, 156 

low  far  may  the  shadow  reach  be- 
yond the  center  of  the  earth  ? 157 

reatest  diameter  of  the  moon's  sha- 
dow where  it  traverses  the  earth,. .     157 
reatest  portion  of  the  earth's  surface 
ever  covered  by  the  moon's  penum- 
bra,      157 

Vloon's  apparent  diameter  compared 
with  the  sun's, 158 

Annular  eclipse,  its  cause, 158 

Direction  in  which  the  eclipse  passes 
on  the  sun's  disk, 159 


ANALYSIS. 


Page 

Greatest  duration  of  total  darkness,. . .     15! 

Eclipses  of  the  sun  more  frequent 
than  of  the  moon,  why  ? 15! 

Lunar  eclipses  oftener  visible,  why  ?     15! 

Radiation  of  light  in  a  total  eclipse  of 
the  sun, 16( 

Interesting  phenomena  of  a  total 
eclipse  of  the  sun, 16( 

Phenomena  of  the  eclipse  of  1806,  de- 
scribed,   16( 

When  does  the  next  total  eclipse  of 
the  sun,  visible  in  the  United 
States,  occur? 16 

Chapter  VIII. — LONGITUDE. — TIDES. 

Objects  of  the  ancients  in  studying 

astronomy, 16 

Ditto  of  the  moderns, 16 

LONGITUDE. — How  to  find  the  differ- 
ence of  longitude  between  two 
places, 161 

Method  by  the  Chronometer  explain- 
ed,   162 

How  to  set  the  chronometer  to  Green- 


wich time, 162  Inferior  and  superior  planets  distin- 


Accuracy  of  some  chronometers, 162 


163 


Longitude  by  eclipses  explained, 

Lunar  method  of  finding  the  longi- 
tude,   

Circumstances     which    render    this 

method  somewhat  difficult, 164 

Disadvantages  of  this  method, 

Degree  of  accuracy  attainable, 165 

TIDES.— defined, 165 

High,  Low,  Spring,  Neap,  Flood,  and 

Ebb  Tide,  severally  defined, 165 

Similar  tides  on  opposite  sides  of  the 

earth, 165 

Interval  between  two  successive  high 

tides...... 165 

Average  height  for  the  whole  globe,     166 

Extreme  height, 166 

Cause  of  the  tides, 166 

Explain  by  figure  56, 166 

Tide-wave  defined, 167 

Comparative  effects  of  the  sun  and 

moon  in  raising  the  tide, 167 

Why  the  moon  raises  a  higher  tide 

than  the  sun, 167 

Springtides  accounted  for, 168 

Neap  tides,         ditto  168 

Power  of  the  sun  or  moon  to  raise 
the  tide,  in  what  ratio  to  its  dis- 
tance,   168 

Influence  of  the  declinations  of  the 

sun  and  moon  on  the  tides, 169 

169 


Page 

Cotidal  Lines  defined, 170 

Derivative  and  Primitive  tides  distin- 
guished,    170 

Velocity  of  the   tide-wave,   circum- 
stances which  affect  it, 171 

Explain  by  figure  59, 171 

Examples  of  very  high  tides, 172 

Unit  of  altitude  defined, 172 

Unit  of  altitude  for  different  places,  172 
Tides  on  the  coast  of  N.  America, 

whence  derived, 173 

Why  no  tides  in  lakes  and  seas, 173 

Intricacy  of  the  problem  of  the  tides,  173 

Atmospheric  tide, 173 

Chapter  IX. — PLANETS — INFERIOR  PLAN- 
ETS, MERCURY  AND  VENUS. 

Etymology  of  the  word  planet, 174 

Planets  known  from  antiquity, 174 

Ditto,  recently  discovered, 174 

Primary  and  Secondary  Planets  dis- 
tinguished,   174 

Whole  number  of  each, 175 

Inclination  of  their  orbits, 175 


guished, 175 


Objections  to  them, 162  Differences  among  the  planets, 175 

-  Distances  from  the  sun  in  miles, 175 

Great    dimensions  of  the   planetary 

orbits, 176 

Flow  long  a  railway  car  would  re- 
quire to  cross  the  orbit  of  Uranus,  176 

Law  of  the  distances, 176 

Mean  distances,  how  determined,....  176 

Diameters  in.  miles, 177 

Flow  ascertained, 177 

Per io  die  t imes, 178 

Which  planets  move  most  rapidly,...  178 
INFERIOR  PLANETS. — Their  proximity 

to  the  sun, 178 

[llustrate  by  figure  60, 179 


When  is  a  planet  said  to  be  in  con- 


junction, 
inferior    and 


179 


superior   conjunctions 
distinguished, 179 

Synodical  revolution  of  a  planet  de- 
fined,      179 

rhy  its  period  exceeds  that  of  the 
planet  in  its  orbit, 179 

To    ascertain    the    synodical   period 

from  the  sidereal, 180 

lynodical   periods   of    Mercury  and 
Venus, 180 

Motions    of    an    inferior  planet   de- 
scribed,      180 

xplain  from  figure  60, 180 


When  is  an  inferior  planet  station- 


ary, 


181 


Explain  from  figures  57  and  58, 

Motion  of  the  tide-wave  not  progres-  Elongation  of  Mercury — when  sta- 

sive, 170      tionary, ". 181 

Tides   of  rivers,   narrow  bays,  how  [Ditto  of  Venus, 181 

produced, 170JPhases  of  Mercury  and  Venus, 182 


lii 


ANALYSIS. 


Page. 
Distance  of  an  inferior  planet  from  Motions  of  the  satellites, 

the  sun,  how  found, 182  Diameter, 193 

Eccentricity  of  the  orbit  of  Mercury,  182  Distances  from  the  primary, 193 

Ditto  of  Venus, 182;Figure  of  their  orbits, 194 

Most  favorable  time  for  determining  Their  inclination  to  the  planet's  equa- 

the  sidereal  revolution  of  a  planet,  183j     tor, 194 

When  is  an  inferior  planet  brightest,  183  Their  eclipses,  how  they  differ  from 
Times  of  their  revolutions  on  their 


axes, 183 

Venus,  her  brightness, 183 

Her  conjunctions  with  the  sun  every 
eight  years, 184 

TRANSITS  OF  THE  INFERIOR  PLANETS 
defined, 184 

Why  a  transit  does  not  occur  at 
every  inferior  conjunction, 184 

Why  the  transits  of  Mercury  occur  in 
May  and  November,  and  those  of 
Venus  in  June  and  December, 185 

Intervals  between  successive  transits, 
how  found, 185 

Why  transits  of  Venus  sometimes  oc- 
cur after  an  interval  of  eight  years,  186 

Why  transits  are  objects  of  so  much 


interest, 186  SATURN,  size 


Method  of  finding  the  sun's  horizontal 

parallax  from  the  transit  of  Venus,  187 
Why  distant  places  are  selected  for 

observing  it, 187 

Explain  the  principle  from  figure  61,  187 
Amount  of  the  sun's  horizontal  par- 
allax,   188 

Indications  of  an  atmosphere  in  Venus,  189 

Whether  Venus  has  any  satellite,... .  189 

Mountains  of  Venus, 189 

Chapter  X. — SUPERIOR  PLANETS,  MARS,  JU- 
PITER, SATURN,  AND  URANUS. 

How  the  superior  planets  are  distin- 
guished from  the  inferior, 189 

MARS,  diameter, 190 

Mean  distance  from  the  sun, 190 


from  those  of  the  moon, 194 

Their  phenomena  explained  from  fig- 
ure 63, 195 

Shadow  seen  traversing  the  disk  of 

the  primary, 196 

Satellite  itself  seen  on  the  disk, 196 

Remarkable    relation    between    the 
mean   motions  of  the   three  first 

satellites, 196 

Consequences  of  this, 196 

Use  of  the  eclipses  of  Jupiter's  satel- 
lites in  finding  the  longitude, 197 

How  it  is  adapted  to  this  purpose,...     197 

Imperfections  of  this  method, 197 

Why  not  practised  at  sea, 198 

Discovery  of  the  progressive  motion 
of  light,  how  made, 198 


199 

Number  of  satellites, 199 

Ring,  double, 199 

Dimensions  of  the  ring  in  several  par- 
ticulars,   199 

Representation  in  the  figure,  (frontis- 
piece,)   199 

Proof  that  the  ring  is  solid  and  opake,  199 
Proof  that  the  axis  of  rotation  is  per- 
pendicular to  the  plane  of  the  ring,  199 

Period  of  rotation, 200 

Compression  of  the  poles, 200 

Peculiar  figure  of  the  planet, 200 

Parallelism  of  the  ring  in  all  parts  of 

its  revolution, 200 

Illustration  by  a    small  disk  and  a 

lamp, 200 

Different  appearances  of  the  ring, 200 

Explain  diagram  64, 201 


Inclination  of  his  orbit, 190  Proof  that  the  ring  shines  by  reflected 


Variation  of  brightness  and  magnitude, 
Explain  the  cause  from  figure  62,.... 


190 
190 

Phases  of  Mars, ~ 191 

Telescopic  appearance, 191 

Revolution  on  his  axis, 191 

Spheroidal  figure, '. 

JUPITER — great  size,  diameter, 191 

Spheroidal  figure, 192 

Rapid  diurnal  revolution, 192 

Distance  from  the  sun, 192 

Periodic  time, 192 

Telescopic  appearance, 192 

Why  astronomers  regard  Jupiter  and 

his  moons  with  so  much  interest,...  192 

Belts,  number,  situation,  cause, 192 

Satellites  of  Jupiter,  number,  situa- 


tion,     193  Irregular  motions,. 


light, 202 

Revolution  of  the  ring, 202 

How  ascertained, 202 

Rings  not  concentric  with  the  planet,  203 

Advantages  of  this  arrangement, 203 

Appearance  of  the  rings  from  the 

planet, 203 

Satellites,  distance  of  the  outermost 

from  the  planet, 204 

Description  of  the  satellites, 204 

URANUS,  distance  and  diameter, 204 

Period  of  revolution,  inclination  of  its 

orbit, . 


205 
205 

His'tory  of  its  discovery, 205 

Satellites,  number,  minuteness, 205 


Appearance  of  the  sun  from  Uranus, 


205 


ANALYSIS. 


Xlii 


Page 

NEW  PLANETS,  their  names, 20f 

Position  in  the  system, 206 

Discovery, 206 

Theoretic  notions  respecting  their  ori- 
gin,   206 

Reason  of  .their  names, 207 

Their  average  distance, 207 

Periodic  Times, 207 

Inclinations  of  their  orbits, 207 

Eccentricity  of       do 20 

Small  size, 208 

Atmospheres, 208 

CHAPTER  XI. — MOTIONS  OF  THE  PLANETARY 
SYSTEM. 

Reasons  for  delaying  the  consideration 
of  the  planetary  motions, 208 

Two  methods  of  studying  the  celestial 
motions, 208 

Notions  of  absolute  space, 209 

Appearance  of  the  planets  from  the 
sun, 209 

Particular  appearance  of  the  orbit  of 
Mercury, 21C 

Mutual  relation  of  the  orbit  of  the 
earth  and  Mercury  considered, 210 

How  the  motions  of  the  other  plan, 
ets  differ  from  from  those  of  Mer- 
cury,   210 

Why  is  the  ecliptic  taken  as  the  stan- 
dard of  reference, 210 

Three  particulars  necessary  in  order 
to  represent  the  actual  positions  of 
the  planetary  orbits, 211 

Why  diagrams  represent  the  orbits  er- 
roneously, as  figure  65, 211 

Inadequate  representations  of  the  so- 
lar system, 213 

How  the  planets  would  be  truly  repre- 
sented,   213 

Two  reasons  why  the  apparent  mo- 
tions are  unlike  the  real, 213 

Explain  figure  66, 214 

Motions  of  Venus  compared  with  those 
of  Mercury, 215 

Apparent  motions  of  the  superior  plan- 
ets, how  far  they  are  like  and  how 
unlike  the  inferior, 215 

Explain  figure  67, 215 

Chapter  XII. — DETERMINATION  OF  THE  PLAN- 
ETARY ORBITS — KEPLER'S  DISCOVERIES — 
ELEMENTS  OF  THE  ORBIT  OF  A  PLANET — 
QUANTITY  OF  MATTER  IN  THE  SUN  AND 
PLANETS — STABILITY  OF  THE  SOLAR  SYS- 
TEM. 

Ptolemy's  views  of  the  figure  of  the 
planetary  orbits, 217 

Kepler's  investigation  of  the  motions 
of  Mars, 218 


Page. 

History  of  the  discovery  Of  Kepler's 
Laws, 218 

Third  Law,  how  modified  by  the  quan- 
tity of  matter, 219 

ELEMENTS,  their  number, 220 

Enumeration  of  them, 220 

Why  we  cannot  find  them  as  we  do 
those  of  the  moon  and  sun, 220 

First  steps  in  the  process, 221 

To  find  the  heliocentric  longitude  and 
latitude  of  a  planet,  figure  68, 221 

To  find  the  position  of  the  nodes,  and 
the  inclination,  figure  69, 222 

To  find  the  periodic  time, 223 

Difficulty  of  finding  when  a  planet  is 
at  itsnode, 223 

Advantage  of  observations  taken  when 
a  planet  is  in  opposition, 223 

Periodic  time,  how  ascertained  most 
accurately, 224 

To  find  the  major  axis  of  the  planet- 
ary orbits, 224 

onstancy  of  the  major  axis, 225 

To  find  the  place  of  the  perihelion, 
figure  71, 226 

To  find  the  place  of  the  planet  in  its 
orbit  at  a  particular  epoch, 227 

To  find  the  eccentricity, -.     227 

QUANTITY  OF  MATTER  IN  THE  SUN 
AND  PLANETS. — How  we  learn  the 
quantity  of  matter  in  a  body, 22b 

Method  by  means  of  the  distances  and 
periodic  times  of  their  satellites, 228 

Vlass  of  the  sun  compared  with  that  of 
the  earth, 229 

Same  result  how  deduced  from  the 
centrifugal  force, 229 

Mass  of  the  planets  that  have  no  sat- 
ellites how  found, 229 

flow  the  quantity  of  matter  in  bodies 

varies 230 

their  densities  vary, 230 

Inferences  from  the  table  of  densities 
and  specific  gravities  of  the  planets,  230 

Perturbations  produced  by  the  planets,    231 

STABILITY  OF  THE  SOLAR  SYSTEM. — 
Probability  of  derangement  in  the 
planetary  motions, 231 

Actual  changes, 231 

Jesuit  of  the  investigations  of  La  Place 
and  La  Grange, 232 

mportant  relation  between  great 
masses  and  small  eccentricities,... .  233 

Chapter  XIII.— COMETS. 

Three  parts  of  a  comet 234 

Description  of  each  part, 234 

Number  of  Comets, 234 

Six  particularly  remarkable 235 

Differences  in  magnitude  and  bright- 

ness, 236 


XIV 


ANALYSIS. 


Page. 

Variations  in  the  same  comet  at  dif- 
ferent returns, 236 

Periods  of  comets, 237 

Distances  of  their  aphelia, 237 

Proof  that  they  shine   by  reflected 

light, 237 

Changes  in  the  tail  at  different  dis- 
tances from  the  sun, 237 

Direction  from  the  sun, 237 

Quantity  of  Matter, 238 

Effect  when  they  pass  very  near  the 

planets, 238 

Proof  that  they  consist  of  matter, —  238 
How  a  comet's  orbit  may  be  entirely 

changed, 239 

How  exemplified  in  the  comet  of  1770,  239 
ORBITS  AND   MOTIONS   OF  COMETS. — 

Nature  of  their  Orbits, 240 

Five  Elements  of  a  Comet, 240 

Investigation  of  these  elements,  why 

so  difficult, 241 

Can  the  length  of  the  major  axis  be 

calculated? 242 

How  determined, 242 

Elements  of  a  comet,  how  calculated  243 
How  a  comet  is  known  to  be  the  same 

as  one  that  has  appeared  before,... .  243 

Exemplified  in  Halley's  comet, 243 

Return  in  1835, 244 

Encke's  comet,   appearance  in  1839,  244 

Proofs  of  a  Resisting  Medium, 244 

Its  consequences, 244 

Physical  nature  of  comets, 245 

How  their   tails  are  supposed  to  be 

formed, 245 

Difficulty  of  accounting  for  the  direc- 
tion of  the  tail, 245 

Supposition  of  Delambre, 246 

Possibility  of  a  comet's  striking  the 

earth, 246 

Instances  of  comets  coming  near  the 

earth, 247 

Consequences  of  a  collision, 247 

Part   III.— OF    THE   FIXED    STARS 
AND  SYSTEM  OF  THE  WORLD. 

Chapter  I. — FIXED  STARS — CONSTELLATIONS. 


Fixed  stars,  why  so  called, 248 

Magnitudes,  how  many  visible  to  the 

naked  eye, 248 

Whole  number  of  magnitudes, 248 

Antiquity  of  the  constellations, 248 

Whether  the  names  are  founded  on 

resemblance, 249 

Names  of  the  individual  stars  of  a  con- 
stellation,   249 

Catalogues  of  the  stars, 249 

Numbers  in  different  catalogues, 249  CLUSTERS 

Utility  of  learning  the  constellations,  250 


Page. 

CONSTELLATIONS. — Aries,  how  recog- 
nized,   251 

Taurus  do. — largest  star  in  Taurus, ...  251 
Gemini, — magnitude  of  Castor,  of  Pol- 
lux,   251 

Cancer,  size  of  its  stars,  Prssepe, . . . .  251 
Leo,  size,  magnitude  of  Regulus,  sickle, 

Denebola, 251 

Virgo,  direction,  Spica,  Vindemiatrix,  252 

Libra,  how  distinguished, 252 

Scorpio,  his  head  how  formed,  An- 

tares,  tail, 252 

Sagittarius,    direction  from   Scorpio, 

how  recognized, 252 

!apricornus,  direction  from  Sagitta- 
rius, t\vo  stars, 

Aquarius,  its  shoulders, ,  252 

Pisces,  situation, 252 

Piscis  Australis,  Fomalhaut, 252 

Andromeda,  how  characterized,  Mi- 

rach,  Almaak, ; 253 

Perseus,  Algol,  Algenib, 253 

Auriga,  situation, — Capella,  its  mag- 
nitude,    253 

Lynx 253 

Leo  Minor,  situation  from  Leo, 253 

oma  Berenices,  direction  from  Leo, 

CorCaroli 253 

Bootes,  Arcturus,  size  and  color, 253 

orona  Borealis,  where  from  Bootes, 

figure, 254 

Hercules,  number  of  stars,  great  extent,  254 

Ophiuchus,  where  from  Hercules, —  254 

Aquila,  three  stars,  Altair,  Antinous,  254 

Delphinus,  four  stars,  tail, 254 

Pegasus,  four  stars  in  a  square,  their 

names, 254 

Ursa  Minor,  Pole-star,  Dipper, 254 

Ursa  Major,  how  recognized,  Point- 
ers, Alioth,  Mizar, 255 

Draco,  position  with  respect  to  the 

Great  and  Little  Bear, 255 

Cepheus,  where  from  the  Dragon,  size 

of  its  stars, 255 

assiopeia's  chair — in  the  Milky  Way,  255 
Cygnus,  where  from  Cassiopeia,  fig- 
ure,   255 

Lyra,  largest  star, 255 

Cetus,  its  extent,  Menkar,  Mira, 256 

Orion,  size  and  beauty,  parts, 256 

Canis  Major,  where  from  Orion,  Sirius,  256 

Canis  Minor,  Procyon, 256 

Hydra — situation — Cor  Hydrae, 256 

"orvus,  how  represented, 256 


hapter  II. — CLUSTERS  OF  STARS — NEBULA 
— VARIABLE  STARS — TEMPORARY  STARS — 
DOUBLE  STARS. 

. — Examples, 257 

Number  of  stars  in  the  Pleiades, 257 


ANALYSIS. 


XV 


Page. 


Stars  of  Coma  Berenices  and  Pree- 
sepe 257 


260  Di 


NEBULA. — Defined, 258 

Examples, 258 

Number  in  Herschel's  Catalogue, —     258 

Herschel's  Views  of  their  nature, 

Figures  of  nebulae, 259 

Nebula  in  the  Sword  of  Orion, 259 

Nebulous  Stars,  defined,  figures, 

Annular  Nebula,  appearance,  exam- 

pie, 

Galaxy   or   Milky  Way,  HerschePs 

views  of  it, 

VARIABLE  STARS. — Defined, 260 

Examples  in  o  Ceti  and  Algol, 261 

TEMPORARY  STARS. — -Defined, 261 

Examples,  why  Hipparchus  number- 
ed the  stars, 261 

Stars  seen  by  the  ancients,  now  miss- 
ing,  .' 261 

DOUBLE  STARS. — Defined,  examples,.    262 
Distance  between  the  double  stars  in 

seconds, 

Colors  of  some  double  stars 262 

Examples, 263 

Number, 263 


Chapter    III. — MOTIONS    OF    THE 
STARS — DISTANCES — NATURE. 


BINARY  STARS,  how  distinguished 
from  common  double  stars, 264 

Examples  of  revolving  stars, 

Inferences  from  the  tabular  view, — 

Particulars  of  y  Virginis, 

Proof  that  the  law  of  gravitation  ex- 
tends to  the  stars, 

Whether  these  stars  are  of  a  planeta- 
ry or  a  cometary  nature, 267 

PROPER  MOTIONS. — Result  on  com- 
paring  the  places  of  certain  stars 
with  those  they  had  in  the  time  of 
Ptolemy, 267 

Conclusion  respecting  the  apparent 
motions  of  certain  stars, 267 

How  the  fact  of  the  sun's  motion 
might  be  proved, 267 

Example  of  stars  having  a  proper  mo- 
tion,   267 

What  class  of  stars  have  the  greatest 
proper  motion? 


Page. 


DISTANCES  OF  THE  FIXED  STARS. — 
What  we  can  determine  respecting 
the  distance  of  the  nearest  star,....  268 

Base  line  for  measuring  this  distance,     269 

Have  the  stars  any  parallax  ? 269 

259  Taking  the  parallax  at  1",  find  the 

distance, 269 

Amount  of  this  distance, 269 

259  Probable     greater    distance    of    the 

smaller  stars, 270 

isputes  respecting  the   parallax   of 

the  stars, 270 

260  To  find  the  parallax  by  means  of  the 

double  stars, 270 

Minuteness  of  the  angles  estimated, .  270 
How  the  magnitude  of  the  stars  is 

affected  by  the  telescope, 271 

NATURE  OF  THE  STARS. — Magnitude 

compared  with  the  earth, 271 

Dr.  Wollaston's  observation  on  their 

comparative  light, 271 

Proofs  that  they  are  suns, 272 

262  Arguments  for  a  Plurality  of  Worlds,  272 


"hapter    IV. — OF    THE    SYSTEM    OF    THE 
WORLD. 


System  of  the  world  defined, 273 

Compared  to  a  machine, 273 

FIXED  Astronomical  knowledge  of  the  an- 
cients,   373 

Things  known  to  Pythagoras, 273 

His  views  of  the  system  of  the  world,  273 

yrstalline  Spheres  of  Eudoxus, 274 

265  Knowledge  possessed  by  Hipparchus,  275 

265  Ptolemaic  System, 276 

265  Deferents  and  epicycles  defined, 276 

Explained  by  figure  74, 276 

266  How  far  this  system  would  explain 

the  phenomena, 277 

Its  absurdities, 277 

Objections  to  the  Ptolemaic  System,.  278 

Tychonic  System, 278 

Its  advantages, 278 

Its  absurdities, 278 

Copernican  System, 279 

Proofs  that  the  earth  revolves, 279 

Ditto,  that  the  planets  revolve  about 

the  sun, 279 

Higher  orders  of  relations  among  the 

stars, 280 

Proofs  of  such  orders, 280 

268  Structure  of  the  material  universe,..  281 


DCr*  Diagrams  for  public  recitations. 


As  many  of  the  figures  of  this  work  are  too  complicated  to  be 
drawn  on  the  black-board  at  each  recitation,  we  have  found  it 
very  convenient  to  provide  a  set  of  permanent  cards  of  paste- 
board, on  which  the  diagrams  are  inscribed  on  so  large  a  scale,  as 
to  be  distinctly  visible  in  all  parts  of  the  lecture  room.  The  let- 
ters may  be  either  made  with  a  pen,  or  better  procured  of  the 
printer,  and  pasted  on. 

The  cards  are  made  by  the  bookbinder,  and  consist  of  a  thick 
paper  board  about  18  by  14  inches,  on  each  side  of  which  a  white 
sheet  is  pasted,  with  a  neat  finish  around  the  edges.  A  loop  at- 
tached to  the  top  is  convenient  for  hanging  the  card  on  a  nail. 

Cards  of  this  description,  containing  diagrams  for  the  whole 
course  of  mathematical  and  philosophical  recitations,  have  been 
provided  in  Yale  College,  and  are  found  a  valuable  part  of  our  ap- 
paratus of  instruction. 

U~p  Several  valuable  articles,  not  contained  in  preceding  edi- 
tions, will  be  found  in  the  Addenda.  Article  IV.,  on  the  Nume- 
rical Relations  existing  between  the  Members  of  the  Solar  System, 
with  problems,  is  particularly  recommended  to  students  in  Astron- 
omy. Notices  of  recent  discoveries  will  be  found  in  Article  V. ; 
and  at  the  end  of  the  volume  is  inserted  a  Syllabus  of  the  Lectures 
on  Astronomy,  delivered  to  the  students  in  Yale  College  after  they 
have  perused  this  treatise. 


INTRODUCTION  TO  ASTRONOMY. 


PRELIMINARY    OBSERVATIONS. 


1.  ASTRONOMY  is  that  science  which  treats  of  the  heavenly  bodies. 

More  particularly,  its  object  is  to  teach  what  is  known  respect- 
ing the  Sun,  Moon,  Planets,  Comets,  and  Fixed  Stars ;  and  also  to 
explain  the  methods  by  which  this  knowledge  is  acquired.  Astron- 
omy is  sometimes  divided  into  Descriptive,  Physical,  and  Practi- 
cal. Descriptive  Astronomy  respects  facts  ;  Physical  Astronomy. 
causes;  Practical  Astronomy,  the  means  of  investigating  the  facts, 
whether  by  instruments,  or  by  calculation.  It  is  the  province  of 
Descriptive  Astronomy  to  observe,  classify,  and  record,  all  the 
phenomena  of  the  heavenly  bodies,  whether  pertaining  to  those 
bodies  individually,  or  resulting  from  their  motions  and  mutual 
relations.  It  is  the  part  of  Physical  Astronomy  to  explain  the 
causes  of  these  phenomena,  by  investigating  and  applying  the 
general  laws  on  which  they  depend  ;  especially  by  tracing  out  all 
the  consequences  of  the  law  of  universal  gravitation.  Practical 
Astronomy  lends  its  aid  to  both  the  other  departments. 

2.  Astronomy  is  the  most  ancient  of  all  the  sciences.     At  a 
period  of  very  high  antiquity,  it  was  cultivated  in  Egypt,  in  Chal- 
dea,  in  China,  and  in  India.     Such  knowledge  of  the  heavenly 
bodies  as  could  be  acquired  by  close  and  long  continued  observa- 
tion, without  the  aid  of  instruments,  was  diligently  amassed ;  and 
tables  of  the  celestial  motions  were  constructed,  which  could  be 
used  in  predicting  eclipses,  and  other  astronomical  phenomena. 

About  500  years  before  the  Christian  era,  Pythagoras,  of 
Greece,  taught  astronomy  at  the  celebrated  school  at  Crotona,  and 
exhibited  more  correct  views  of  the  nature  of  the  celestial  mo- 
tions, than  were  entertained  by  any  other  astronomer  of  the  an- 
cient world.  His  views,  however,  were  not  generally  adopted, 

1 


2  PRELIMINARY  OBSERVATIONS. 

but  lay  neglected  for  nearly  2000  years,  when  they  were  revived 
and  established  by  Copernicus  and  Galileo.  The  most  celebrated 
astronomical  school  of  antiquity  was  at  Alexandria,  in  Egypt, 
which  was  established  and  sustained  by  the  Ptolemies,  (Egyptian 
princes,)  about  300  years  before  the  Christian  era.  The  employ- 
ment of  instruments  for  measuring  angles,  and  the  introduction  of 
trigonometrical  calculations  to  aid  the  naked  powers  of  observa- 
tion, gave  to  the  Alexandrian  astronomers  great  advantages  over 
all  their  predecessors.  The  most  able  astronomer  of  the  Alexan- 
drian school  was  Hipparchus,  who  was  distinguished  above  all  the 
ancients  for  the  accuracy  of  his  astronomical  measurements  and 
determinations.  The  knowledge  of  astronomy  possessed  by  the 
Alexandrian  school,  and  recorded  in  the  Almagest*  or  great  work 
of  Ptolemy,  constituted  the  chief  of  what  was  known  of  our 
science  during  the  middle  ages,  until  the  fifteenth  and  sixteenth 
centuries,  when  the  labors  of  Copernicus  of  Prussia,  Tycho  Brake 
of  Denmark,  Kepler  of  Germany,  and  Galileo  of  Italy,  laid  the 
solid  foundations  of  modern  astronomy.  Copernicus  expounded 
the  true  theory  of  the  celestial  motions ;  Tycho  Brahe  carried 
the  use  of  instruments  and  the  art  of  astronomical  observation  to 
a  far  higher  degree  of  accuracy  than  had  ever  been  done  before  ; 
Kepler  discovered  the  great  laws  of  the  planetary  motions  ;  and 
Galileo,  having  first  enjoyed  the  aid  of  the  telescope,  made  innu- 
merable discoveries  in  the  solar  system.  Near  the  beginning  of 
the  eighteenth  century,  Sir  Isaac  Newton  discovered,  in  the  law 
of  universal  gravitation,  the  great  principle  that  governs  the  ce- 
lestial motions  ;  and  recently,  La  Place  has  more  fully  completed 
what  Newton  began,  having  followed  out  all  the  consequences  of 
the  law  of  universal  gravitation,  in  his  great  work,  the  Mecan- 
ique  Celeste. 

X 

3.  Among  the  ancients,  astronomy  was  studied  chiefly  as  sub- 
sidiary to  astrology.  ASTROLOGY  was  the  art  of  divining  future 
events  by  the  stars.  It  was  of  two  kinds,  natural  and  judicial. 
Natural  Astrology,  aimed  at  predicting  remarkable  occurrences  in 
the  natural  world,  as  earthquakes,  volcanoes,  tempests,  and  pesti- 
lential diseases.  Judicial  Astrology,  aimed  at  foretelling  the  fates 
of  individuals,  or  of  empires. 


PRELIMINARY    OBSERVATIONS.  3 

4.  Astronomers  of  every  age,  have  been  distinguished  for  their 
persevering  industry,  and  their  great  love  of  accuracy.     They 
have  uniformly  aspired  to  an  exactness  in  their  inquiries,  far  be- 
yond what  is  aimed  at  in  most  geographical  investigations,  satis- 
fied with  nothing  short  of  numerical  accuracy,  wherever  this  is 
attainable ;  and  years  of  toilsome  observation,  or  laborious  calcu- 
lation, have  been  spent  with  the  hope  of  attaining  a  few  seconds 
nearer  to  the  truth.     Moreover,  a  severe  but  delightful  labor  is 
imposed  on  all  who  would  arrive  at  a  clear  and  satisfactory  knowl- 
edge of  the  subject  of  astronomy.     Diagrams,  artificial  globes, 
orreries,  and  familiar  comparisons  and  illustrations,  proposed  by 
the  author  or  the  instructor,  may  afford  essential  aid  to  the  learner, 
but  nothing  can  convey  to  him  a  perfect  comprehension  of  the 
celestial  motions,  without  much  diligent  study  and  reflection. 

5.  In  expounding  the  doctrines  of  astronomy,  we  do  not,  as  in 
geometry,  claim  that  every  thing  shall  be  proved  as  soon  as  as- 
serted.    We  may  first  put  the  learner  in  possession  of  the  leading 
facts  of  the  science,  and  afterwards  explain  to  him  the  methods 
by  which  those  facts  were  discovered,  and  by  which  they  may 
be  verified ;  we  may  assume  the  principles  of  the  true  system  of 
the  world,  and  employ  those  principles  in  the  explanation  of  many 
subordinate  phenomena,  while  we  reserve  the  discussion  of  the 
merits  of  the  system  itself,  until  the  learner  is  extensively  ac- 
quainted with  astronomical  facts,  and  therefore  better  able  to  ap- 
preciate the  evidence  by  which  the  system  is  established. 

6.  The  Copernican  System  is  that  which  is  held  to  be  the  true 
system  of  the  world.     It  maintains  (1,)  That  the  apparent  diur- 
nal revolution  of  the  heavenly  bodies,  from  east  to  west,  is  owing 
to  the  real  revolution  of  the  earth  on  its  own  axis  from  west  to 
east,  in  the  same  time ;  and  (2,)  That  the  sun  is  the  center  around 
which  the  parth  and  planets  all  revolve  from  west  to  east,  con- 
trary to  the  opinion  that  the  earth  is  the  center  of  motion  of  the 
sun  and  planets. 

7.  We  shall  treat,  first,  of  the  Earth  in  its  astronomical  rela- 
tions ;  secondly,  of  the  Solar  System ;  and,  thirdly,  of  the  Fixed 
Stars. 


PART  I. — OF  THE  EARTH 


CHAPTER  I. 


OP  THE  FIGURE  AND  DIMENSIONS  OF  THE  EARTH,  AND  THE  DOCTRINE 
OF  THE  SPHERE. 

8.  The  figure  of  the  earth  is  nearly  globular.     This  fact  is 
known,  first,  by  the  circular  form  of  its  shadow  cast  upon  the 
moon  in  a  lunar  eclipse ;  secondly,  from  analogy,  each  of  the 
other  planets  being  seen  to  be  spherical ;  thirdly,  by  our  seeing 
the  tops  of  distant  objects  while  the  other  parts  are  invisible,  as 
the  topmast  of  a  ship,  while  either  leaving  or  approaching  the 
shore,  or  the  lantern  of  a  light-house,  which,  when  first  descried 
at  a  distance  at  sea,  appears  to  glimmer  upon  the  very  surface  of 
the  water  ;  fourthly,  by  the  depression  or  dip  of  the  horizon  when 
the  spectator  is  on  an  eminence  ;  and,  finally,  by  actual  observa- 
tions and  measurements,  made  for  the  express  purpose  of  ascer- 
taining the  figure  of  the  earth,  by  means  of  which  astronomers  are 
enabled  to  compute  the  distances  from 

the  center  of  the  earth  of  various  places 
on  its  surface,  which  distances  are  found 
to  be  nearly  equal. 

9.  The  Dip  of  the  Horizon,  is  the  ap- 
parent angular  depression  of  the  hori- 
zon, to  a  spectator  elevated  above  the 
general  level  of  the  earth.      The  eye 
thus  situated,  takes  in  more  than  a  ce- 
lestial hemisphere,  the  excess  being  the 
measure  of  the  dip. 

Thus,  in  Fig.  1,  let  AO  represent  the 


FIGURE  AND  DIMENSIONS.  5 

height  of  a  mountain,  ZO  the  direction  of  the  plumb  line,  HOR  a 
line  touching  the  earth  at  the  point  O,'  and  at  right  angles  to  the 
plumb  line,  C  the  center  of  the  earth,  DAE  the  portion  of  the 
earth's  surface  seen  from  O;  OD,  OE,  lines  drawn  from  the 
place  of  the  spectator  to  the  most  distant  parts  of  the  horizon, 
and  CD  a  radius  of  the  earth.  The  dip  of  the  horizon  is  the  an- 
gle HOD  or  ROE.  Now  the  angle  made  between  the  direction 
of  the  plumb  line  and  that  of  the  extreme  line  of  the  horizon  or 
the  surface  of  the  sea,  namely,  the  angle  ZOD,  can  be  easily 
measured ;  and  subtracting  the  right  angle  ZOH  from  ZOD,  the 
remainder  is  the  dip  of  the  horizon,  from  which  the  length  of  the 
line  OD  may  be  calculated,  (see  Art.  10,)  the  height  of  the  spec- 
tator, that  is,  the  line  OA,  being  known.  This  length,  to  whatever 
point  of  the  horizon  the  line  is  drawn,  is  always  found  to  be  the 
same ;  and  hence  it  is  inferred,  that  the  boundary  which  limits 
the  view  on  all  sides,  is  a  circle.  Moreover,  at  whatever  elevation 
the  dip  of  the  horizon  is  taken,  in  any  part  of  the  earth,  the 
space  seen  by  the  spectator  is  always  circular.  Hence  the  sur- 
face of  the  earth  is  spherical. 

10.  The  earth  being  a  sphere,  the  dip  of  the  horizon  HOD= 
OCD.  Therefore,  to  find  the  dip  of  the  horizon  corresponding 
to  any  given  height  AO*  (the  diameter  of  the  earth  being  known,) 
we  have  in  the  triangle  OCD,  the  right  angle  at  D,  and  the  two 
sides  CD,  CO,  to  find  the  angle  OCD.  Therefore, 

CO  :  rad. : :  CD  :  cos.  OCD.  Learning  the  dip  corresponding 
to  different  altitudes,  by  giving  to  the  line  AO  different  values, 
we  may  arrange  the  results  in  a  table. 

*  The  learner  will  remark  that  the  line  AO,  as  drawn  in  the  figure,  is  much  larger 
m  proportion  to  CA  than  is  actually  the  case,  and  that  the  angle  HOD  is  much  too 
great  for  the  reality.  Such  disproportions  are  very  frequent  in  astronomical  diagrams, 
especially  when  some  of  the  parts  are  exceedingly  small  compared  with  others ;  and 
hence  the  diagrams  employed  in  astronomy  are  not  to  be  regarded  as  true  pictures  of 
the  magnitudes  concerned,  but  merely  as  representing  their  abstract  geometrical  re. 
lations. 


THE   EARTH. 


Table  showing  the  Dip  of  the  Horizon  at  different  elevations,  from 
I  foot  to  WO  feet* 


Feet. 

/    // 

Feet. 

/    // 

Feet. 

/    // 

1 

0.59 

13 

3.33 

26 

5.01 

2 

1.24 

14 

3.41 

28 

5.13 

3 

1.42 

15 

3.49 

30 

5.23 

4 

1.58 

16 

3.56 

35 

5.49 

5 

2.12 

17 

4.03 

40 

6.14 

6 

2.25 

18 

4.11 

45 

6.36 

7 

2.36 

19 

4.17 

50 

6.58 

8 

2.47 

20 

4.24 

60 

7.37 

9 

2.57 

21 

4.31 

70 

8.14 

10 

3.07 

22 

4.37 

80 

8.48 

11 

3.16 

23 

4.43 

90 

9.20 

12 

3.25 

24 

4.49 

100 

9.51 

Such  a  table  is  of  use  in  estimating  the  altitude  of  a  body 
above  the  horizon,  when  the  instrument  (as  usually  happens)  is 
more  or  less  elevated  above  the  general  level  of  the  earth.  For 
if  it  is  a  star  whose  altitude  above  the  horizon  is  required,  the 
instrument  being  situated  at  O,  (Fig.  1,)  the  inquiry  is  how  far 
the  star  is  elevated  above  the  level  HOR,  but  the  angle  taken  is 
that  above  the  visible  horizon  OD.  The  dip,  therefore,  or  the 
angle  HOD,  corresponding  to  the  height  of  the  point  O,  must  be 
subtracted,  to  obtain  the  true  altitude.  On  the  Peak  of  Tene- 
rifte,  a  mountain  13,000  feet  high,  Humboldt  observed  the  surface 
of  the  sea  to  be  depressed  on  all  sides  nearly  2  degrees.  The 
sun  arose  to  him  12  minutes  sooner  than  to  an  inhabitant  of  the 
plain ;  and  from  the  plain,  the  top  of  the  mountain  appeared  en- 
lightened 12  minutes  before  the  rising  or  after  the  setting  of 
the  sun. 

11.  The  foregoing  considerations  show  that  the  form  of  the 
earth  is  spherical ;  but  more  exact  determinations  prove,  that  the 
earth,  though  nearly  globular,  is  not  exactly  so  :  its  diameter  from 
the  north  to  the  south  pole  is  about  26  miles  less  than  through 
the  equator,  giving  to  the  earth  the  form  of  an  oblate  spheroid,! 

*  This  table  includes  the  allowance  for  refraction. 

t  An  oblate  spheroid  is  the  solid  described  by  the  revolution  of  an  ellipse  about  its 
shorter  axis. 


FIGURE  AND  DIMENSIONS.  7 

or  a  flattened  sphere  resembling  an  orange.  We  shall  reserve  the 
explanations  of  the  methods  by  which  this  fact  is  established, 
until  the  learner  is  better  prepared  than  at  present  to  understand 
them. 

12.  The  mean  or  average  diameter  of  the  earth,  is  7912^4  miles, 
a  measure  which  the  learner  should  fix  in  his  memory  as  a  stand- 
ard of  comparison  in  astronomy,  and  of  which  he  should  endeavor 
to  form  the  most  adequate  conception  in  his  power.  The  circum- 
ference of  the  earth  is  about  25,000  miles  (24857.5).*  Although 
the  surface  of  the  earth  is  uneven,  sometimes  rising  in  high  moun- 
tains, and  sometimes  descending  in  deep  valleys,  yet  these  eleva- 
tions and  depressions  are  so  small  in  comparison  with  the  immense 
volume  of  the  globe,  as  hardly  to  occasion  any  sensible  deviation 
from  a  surface  uniformly  curvilinear.  The  irregularities  of  the 
earth's  surface  in  this  view,  are  no  greater  than  the  rough  points 
on  the  rind  of  an  orange,  which  do  not  perceptibly  interrupt  its 
continuity  ;  for  the  highest  mountain  on  the  globe  is  only  about 
five  miles  above  the  general  level  ;  and  the  deepest  mine  hitherto 

opened  is  only  about  half  a  mile.f    Now  i^r^r^;'  or  about 


one  sixteen  hundredth  part  of  the  whole  diameter,  an  inequality 
which,  in  an  artificial  globe  of  eighteen  inches  diameter,  amounts 
to  only  the  eighty-eighth  part  of  an  inch. 

13.  The  diameter  of  the  earth,  con- 
sidered as  a  perfect  sphere,  may  be  de- 
termined by  means  of  observations  on 
a  mountain  of  known  elevation,  seen 
in  the  horizon  from  the  sea.  Let  BD 
(Fig.  2,)  be  a  mountain  of  known 
height  a,  whose  top  is  seen  in  the  hori- 
zon by  a  spectator  at  A,  b  miles  from  it. 
Let  x  denote  the  radius  of  the  earth. 
Then  &  +  b2  =  (x+a)*  =  &  +  2ax  +  a?. 

*  It  will  generally  be  sufficient  to  treasure  up  in  the  memory  the  round  number, 
but  sometimes,  in  astronomical  calculations,  the  more  exact  number  may  be  required, 
and  it  is  therefore  inserted. 

t  Sir  John  HerscheL 


8  THE   EARTH. 

7,2         2 

Hence,  2ax=tf*—cP,  and  x=—  -  —  .      For  example,  suppose   the 

height  of  the  mountain  is  just  one  mile  ;  then  it  will  be  found, 
by  observation,  to  be  visible  on  the  horizon  at  the  distance  of 


89  miles=6.      Hence,     z_-=:=3960=  radius 

Zo,  &  2 

of  the  earth,  and  7920=the  earth's  diameter. 

14.  Another  method,  and  the  most  ancient,  is  to  ascertain  the 
distance  on  the  surface  of  the  earth,  corresponding  to  a  degree  of 
latitude.  Let  us  select  two  convenient  places,  one  lying  directly 
north  of  the  other,  whose  difference  of  latitude  is  known.  Sup- 
pose this  difference  to  be  1°  30',  and  the  distance  between  the 
two  places,  as  measured  by  a  chain,  to  be  104  miles.  Then, 
since  there  are  360  degrees  of  latitude  in  the  entire  circumference, 


1°  30'  :  104  :  :  360°  :  24960.    And  —       =7944. 


The  foregoing  approximations  are  sufficient  to  show  that  the 
earth  is  about  8,000  miles  in  diameter. 

15.  The  greatest  difficulty  in  the  way  of  acquiring  correct 
views  in  astronomy,  arises  from  the  erroneous  notions  that  pre- 
occupy the  mind.  To  divest  himself  of  these,  the  learner  should 
conceive  of  the  earth  as  a  huge  globe  occupying  a  small  portion 

Fig.  3. 


DOCTRINE  OF  THE    SPHERE.  0 

of  space,  and  encircled  on  all  sides  with  the  starry  sphere.  He 
should  free  his  mind  from  its  habitual  proneness  to  consider  one 
part  of  space  as  naturally  up  and  another  down,  and  view  him- 
self as  subject  to  a  force  which  binds  him  to  the  earth  as  truly  as 
though  he  were  fastened  to  it  by  some  invisible  cords  or  wires,  as 
the  needle  attaches  itself  to  all  sides  of  a  spherical  loadstone.  He 
should  dwell  on  this  point  until  it  appears  to  him  as  truly  up  in 
the  direction  of  BB ,  CC  ,  DD ,  (Fig.  3,)  when  he  is  at  B,  C,  and 
D,  respectively,  as  in  the  direction  of  AA  when  he  is  at  A. 

DOCTRINE  OF   THE   SPHERE. 

16.  The  definitions  of  the  different  lines,  points,  and  circles, 
which  are  used  in  astronomy,  and  the  propositions  founded  upon 
them,  compose  the  Doctrine  of  the  Sphere.* 

17.  A  section  of  a  sphere  by  a  plane  cutting  it  in  any  manner, 
is  a  circle.     Great  circles  are  those  which  pass  through  the  center 
of  the  sphere,  and  divide  it  into  two  equal  hemispheres :  Small 
circles,  are  such  as  do  not  pass  through  the  center,  but  divide  the 
sphere  into  two  unequal  parts.     Every  circle,  whether  great  or 
small,  is  divided  into  360  equal  parts  called  degrees.     A  degree, 
therefore,  is  not  any  fixed  or  definite  quantity,  but  only  a  certain 
aliquot  part  of  any  circle. 

18.  The  Axis  of  a  circle,  is  a  straight  line  passing  through  its 
center  at  right  angles  to  its  plane. 

19.  The  Pole  of  a  great  circle,  is  the  point  on  the  sphere  where 
its  axis  cuts  through   the  sphere.     Every  great  circle  has  two 
poles,  each  of  which  is  every  where  90°  from  the  great  circle. 
For,  the  measure  of  an  angle  at  the  center  of  a  sphere,  is  the 
arc  of  a  great  circle  intercepted  between  the  two  lines  that  con- 
tain the  angle ;  and,  since  the  angle  made  by  the  axis  and  any 
radius  of  the  circle  is  a  right  angle,  consequently  its  measure  on 
the  sphere,  namely,  the  distance  fropa  the  pole  to  the  circumfer- 

*  It  is  presumed  that  many  of  those  who  read  this  work,  will  have  studied  Spherical 
Geometry ;  but  it  is  so  important  to  the  student  of  astronomy  to  have  a  clear  idea  of 
the  circles  of  the  sphere,  that  it  is  thought  best  to  introduce  them  here. 


10  THE   EARTH. 

ence  of  the  circle,  must  be  90°.  If  two  great  circles  cut  each 
other  at  right  angles,  the  poles  of  each  circle  lie  in  the  circum- 
ference of  the  other  circle.  For  each  circle  passes  through  the 
axis  of  the  other. 

20.  All  great  circles  of  the  sphere  cut  each  other  in  two  points 
diametrically  opposite,  and  consequently,  their  points  of  section 
are  180°  apart.     For  the  line  of  common  section,  is  a  diameter 
in  both  circles,  and  therefore  bisects  both. 

21.  A  great  circle  which  passes  through  the  pole  of  another 
great  circle,  cuts  the  latter  at  right  angles.     For,  since  it  passes 
through  the  pole  and  the  center  of  the  circle,  it  must  pass  through 
the  axis ;  which  being  at  right  angles  to  the  plane  of  the  circle, 
every  plane  which  passes  through  it  is  at  right  angles  to  the  same 
plane. 

The  great  circle  which  passes  through  the  pole  of  another  great 
circle  and  is  at  right  angles  to  it,  is  called  a  secondary  to  that  circle. 

22.  The  angle  made  by  two  great  circles  on  the  surface  of  the 
sphere,  is  measured  by  the  arc  of  another  great  circle,  of  which 
the  angular  point  is  the  pole,  being  the  arc  of  that  great  circle 
intercepted  between  those  two  circles.     For  this  arc  is  the  meas- 
ure of  the  angle  formed  at  the  center  of  the  sphere  by  two  radii, 
drawn  at  right  angles  to  the  line  of  common  section  of  the  two 
circles,  one  in  one  plane  and  the  other  in  the  other,  which  angle 
is  therefore  that  of  the  inclination  of  those  planes. 

23.  In  order  to  fix  the  position  of  any  plane,  either  on  the  sur- 
face of  the  earth  or  in  the  heavens,  both  the  earth  and  the  heav- 
ens are  conceived  to  be  divided  into  separate  portions  by  circles, 
which  are  imagined  to  cut  through  them  in  various  ways.     The 
earth  thus  intersected  is  called  the  terrestrial,  and  the  heavens  the 
celestial  sphere.     The  learner  will  remark,  that  these  circles  have 
no  existence  in  nature,  but  are  mere  landmarks,  artificially  con- 
trived for  convenience  of  reference.     On  account  of  the  immense 
distance  of  the  heavenly  bodies,  they  appear  to  us,  wherever  we 
are  placed,  to  be  fixed  in  the  same  concave  surface,  or  celestial 


DOCTRINE    OF   THE    SPHERE.  11 

vault.  The  great  circles  of  the  globe,  extended  every  way  to 
meet  the  concave  surface  of  the  heavens,  become  circles  of  the 
celestial  sphere. 

'$" 

24.  The  Horizon  is  the  great  circle  which  divides  the  earth 
into  upper  and  lower  hemispheres,  and  separates  the  visible  heav- 
ens from  the  invisible.     This  is  the  rational  horizon.     The  sen- 
sible horizon,  is  a  circle  touching  the  earth  at  the  place  of  the 
spectator,  and  is  bounded  by  the  line  in  which  the  earth  and  skies 
seem  to  meet.     The  sensible  horizon  is  parallel  to  the  rational, 
but  is  distant  from  it  by  the  semi-diameter  of  the  earth,  or  nearly 
4,000  miles.     Still,  so  vast  is  the  distance  of  the  starry  sphere, 
that  both  these  planes  appear  to  cut  that  sphere  in  the  same  line ; 
so  that  we  see  the  same  hemisphere  of  stars  that  we  should  see  if 
the  upper  half  of  the  earth  were  removed,  and  we  stood  on  the 
rational  horizon. 

25.  The  poles  of  the  horizon  are  the  zenith  and  nadir.     The 
Zenith  is  the  point  directly  over  our  head,  and  the  Nadir  that  di- 
rectly under  our  feet.     The  plumb  line  is  in  the  axis  of  the  hori- 
zon, and  consequently  directed  towards  its  poles. 

Every  place  on  the  surface  of  the  earth  has  its  own  horizon ; 
and  the  traveller  has  a  new  horizon  at  every  step,  always  extend- 
ing 90  degrees  from  his  zenith  in  all  directions. 

26.  Vertical  circles  are  those  which  pass  through  the  poles  of 
the  horizon,  perpendicular  to  it. 

The  Meridian  is  that  vertical  circle  which  passes  through  the 
north  and  south  points. 

The  Prime  Vertical,  is  that  vertical  circle  which  passes  through 
the  east  and  west  points. 

27.  As  in  geometry,  we  determine  the  position  of  any  point  by 
means  of  rectangular  coordinates,  or  perpendiculars  drawn  from 
the  point  to  planes  at  right  angles  to  each  other,  so  in  astron- 
omy we  ascertain  the  place  of  a  body,  as  a  fixed  star,  by  taking 
its  angular  distance  from  two  great  circles,  one  of  which  is  per- 
pendicular to  the  other.    Thus  the  horizon  and  the  meridian,  or  the 


12  THE    EARTH. 

horizon  and  the  prime  vertical,  are  coordinate  circles  used  for  such 
measurements. 

The  Altitude  of  a  body,  is  its  elevation  above  the  horizon  meas- 
ured on  a  vertical  circle. 

The  Azimuth  of  a  body,  is  its  distance  measured  on  the  hori- 
zon from  the  meridian  to  a  vertical  circle  passing  through  the  body. 

The  Amplitude  of  a  body,  is  its  distance  on  the  horizon,  from 
the  prime  vertical,  to  a  vertical  circle  passing  through  the  body. 

Azimuth  is  reckoned  90°  from  either  the  north  or  south  point ; 
and  amplitude  90°  from  either  the  east  or  west  point.  Azimuth 
and  amplitude  are  mutually  complements  of  each  other.  When  a 
point  is  on  the  horizon,  it  is  only  necessary  to  count  the  number 
of  degrees  of  the  horizon  between  that  point  and  the  meridian, 
in  order  to  find  its  azimuth ;  but  if  the  point  is  above  the  horizon, 
then  its  azimuth  is  estimated  by  passing  a  vertical  circle  through 
it,  and  reckoning  the  azimuth  from  the  point  where  this  circle  cuts 
the  horizon. 

The  Zenith  Distance  of  a  body  is  measured  on  a  vertical  cir- 
cle, passing  through  that  body.  It  is  the  complement  of  the  alti- 
tude. 

-28.  The  Axis  of  the  Earth  is  the  diameter,  on  which  the  earth 
is  conceived  to  turn  in  its  diurnal  revolution.  The  same  line  con- 
tinued until  it  meets  the  starry  concave,  constitutes  the  axis  of  the 
celestial  sphere. 

The  Poles  of  the  Earth  are  the  extremities  of  the  earth's  axis: 
the  Poles  of  the  Heavens,  the  extremities  of  the  celestial  axis. 

29.  The  Equator  is  a  great  circle  cutting  the  axis  of  the  earth 
at  right  angles.     Hence  the  axis  of  the  earth  is  the  axis  of  the 
equator,  and  its  poles  are  the  poles  of  the  equator.     The  intersec- 
tion of  the  plane  of  the  equator  with  the  surface  of  the  earth, 
constitutes  the  terrestrial,  and  with  the  concave  sphere  of  the 
heavens,  the  celestial  equator.     The  latter,  by  way  of  distinction, 
is  sometimes  denominated  the  equinoctial. 

30.  The  secondaries  to  the  equator,  that  is,  the  great  circles 
passing  through  the  poles  of  the  equator,  are  called  Meridians 


DOCTRINE  OF    THE    SPHERE.  13 

because  that  secondary  which  passes  through  the  zenith  of  any 
place  is  the  meridian  of  that  place,  and  is  at  right  angles  both  to 
the  equator  and  the  horizon,  passing  as  it  does  through  the  poles 
of  both.  (Art.  21.)  These  secondaries  are  also  called  Hour  Circles, 
because  the  arcs  of  the  equator  intercepted  between  them  are  used 
as  measures  of  time. 

31.  The  Latitude  of  a  place  on  the  earth,  is  its  distance  from 
the  equator  north  or  south.     The  Polar  Distance,  or  angular  dis- 
tance from  the  nearest  pole,  is  the  complement  of  the  latitude. 

32.  The  Longitude  of  a  place  is  its  distance  from  some  stand- 
ard meridian,  either  east  or  west,  measured  on  the  equator.     The 
meridian  usually  taken  as  the  standard,  is  that  of  the  Observatory 
of  Greenwich,  near  London.     If  a  place  is  directly  on  the  equator, 
we  have  only  to  inquire  how  many  degrees  of  the  equator  there 
are  between  that  place  and  the  point  where  the  meridian  of  Green- 
wich cuts  the  equator.     If  the  place  is  north  or  south  of  the  equa- 
tor, then  its  longitude  is  the  arc  of  the  equator  intercepted  between 
the  meridian  which  passes  through  the  place,  and  the  meridian  ot 
Greenwich. 

33.  The  Ecliptic  is  a  great  circle  in  which  the  earth  perforate 
its  annual  revolution  around  the  sun.     It  passes  through  the  center 
of  the  earth  and  the  center  of  the  sun.     It  is  found  by  observa- 
tion that  the  earth  does  not  lie  with  its  axis  at  right  angles  to  the 
plane  of  the  ecliptic,  but  that  it  is  turned  about  23£  degrees  out  of 
a  perpendicular  direction,  making  an  angle  with  the  plane  itself  of 
66£°.     The  equator,  therefore,  must  be  turned  the  same  distance 
out  of  a  coincidence  with  the  ecliptic,  the  two  circles  making 
an  angle  with  each  other  of  23J°,  (23°  27'  40".)     It  is  particu- 
larly important  for  the  learner  to  form  correct  ideas  of  the  eclip- 
tic, and  of  its  relations  to  the  equator,  since  to  these  two  circles  a 
great  number  of  astronomical  measurements  and  phenomena  are 
referred. 

34.  The  Equinoctial  Points,  or  Equinoxes*  are  the  intersec- 

*  The  term  Equinoxes  strictly  denotes  the  times  when  the  sun  arrives  at  the  equi. 
noctial  points,  but  it  is  also  frequently  used  to  denote  those  points  themselves. 


14  THE    EARTH. 

tions  of  the  ecliptic  and  equator.  The  time  when  the  sun  crosses 
the  equator  in  returning  northward  is  called  the  vernal,  and  in 
going  southward,  the  autumnal  equinox.  The  vernal  equinox 
occurs  about  the  21st  of  March,  and  the  autumnal  the  22d  of 
September. 

35.  The  Solstitial  Points  are  the  two   points  of  the  ecliptic 
most  distant  from  the  equator.     The  times  when  the  sun  comes 
to  them  are  called  solstices.     The  summer  solstice  occurs  about 
the  22d  of  June,  and  the  winter  solstice  about  the  22d  of  De- 
cember. 

The  ecliptic  is  divided  into  twelve  equal  parts  of  30°  each, 
called  signs,  which,  beginning  at  the  vernal  equinox,  succeed  each 
other  in  the  following  order : 

Northern.  Southern. 

1.  Aries       T  7.  Libra             ^ 

2.  Taurus    8  8.  Scorpio         tn, 

3.  Gemini  n  9.  Sagittarius     / 

4.  Cancer  S£  10.  Capricornus  V3 

5.  Leo         £1  11.  Aquarius       ~ 

6.  Virgo     nj;  12,  Pisces            x 

The  mode  of  reckoning  on  the  ecliptic,  is  by  signs,  degrees, 
minutes,  and  seconds.  The  sign  is  denoted  either  by  its  name 
or  its  number.  Thus  100°  may  be  expressed  either  as  the  10th 
degree  of  Cancer,  or  as  3s  10°. 

36.  Of  the  various  meridians,  two  are  distinguished  by  the 
name  of  Colures.     The  Equinoctial  Colure,  is  the  meridian  which 
passes  through  the  equinoctial  points.     The  Solstitial  Colure,  is 
the  meridian  which  passes  through  the  solstitial  points.     As  the 
solstitial  points  are  90°  from  the  equinoctial  points,  so  the  sol- 
stitial colure  is  90°  from  the  equinoctial  colure.     It  is  also  at  right 
angles,  or  a  secondary  to  both  the  ecliptic  and  equator.     For  like 
every  other  meridian,  it  is  of  course  perpendicular  to  the  equator, 
passing  through  its  poles.     Moreover,  the  equinox,  being  a  point 
both  in  the  equator  and  in  the  ecliptic,  is  90°  from  the  solstice, 
from  the  pole  of  the  equator,  and  from  the  pole  of  the  ecliptic. 


DOCTRINE    OF    THE    SPHERE. 


15 


Hence  the  solstitial  colure,  which  passes  through  the  solstice  and 
the  pole  of  the  equator,  passes  also  through  the  pole  of  the  ecliptic, 
being  the  great  circle  of  which  the  equinox  itself  is  the  pole. 
Consequently,  the  solstitial  colure  is  a  secondary  to  both  the  equa- 
tor and  the  ecliptic.  (See  Arts.  19,  20,  21.) 

37.  The  position  of  a  celestial  body  is  referred  to  the  equator 
by  its  right  ascension  and  declination.  (See  Art.  27.)  Right 
Ascension,  is  the  angular  distance  from  the  vernal  equinox,  meas- 
ured on  the  equator.  If  a  star  is  situated  on  the  equator,  then  its 
right  ascension  is  the  number  of  degrees  of  the  equator  between 
the  star  and  the  vernal  equinox.  But  if  the  star  is  north  or  south 
of  the  equator,  then  its  right  ascension  is  the  arc  of  the  equator 
intercepted  between  the  vernal  equinox  and  that  secondary  to  the 
equator  which  passes  through  the  star.  Declination  is  the  dis- 
tance of  a  body  from  the  equator,  measured  on  a  secondary  to  the 
latter.  Therefore,  right  ascension  and  declination  correspond  to 
terrestrial  longitude  and  latitude,  right  ascension  being  reckoned 
from  the  equinoctial  colure,  in  the  same  manner  as  longitude  is 
reckoned  from  the  meridian  of  Greenwich.  On  the  other  hand, 
celestial  longitude  and  latitude  are  referred,  not  to  the  equator, 
but  to  the  ecliptic.  Celestial  Longitude,  is  the  distance  of  a  body 
from  the  vernal  equinox  reckoned  on  the  ecliptic.  Celestial  Lati- 
tude, is  distance  from  the  ecliptic  measured  on  a  secondary  to  the 
latter.  Or,  more  briefly,  Longitude  is  distance  on  the  ecliptic  ; 
Latitude,  distance  from  the  ecliptic.  The  North  Polar  Distance 
of  a  star,  is  the  complement  of  its  declination. 


38.  Parallels  of  Latitude  are  small 
circles  parallel  to  the  equator.  They 
constantly  diminish  in  size  as  we  go 
from  the  equator  to  the  pole,  the  ra- 
dius being  always  equal  to  the  cosine 
of  the  latitude.  In  fig.  4,  let  HO  be 
the  horizon,  EQ  the  equator,  PP  the 
axis  of  the  earth,  ZN  the  prime  ver- 
tical, and  ZL  a  parallel  of  latitude  of 
any  place  Z.  Then  ZE  is  the  lati- 


16  THE    EARTH. 

tude,  (Art.  31,)  and  ZP  the  complement  of  the  latitude ;  but  Zn 
the  radius  of  the  parallel  of  latitude  ZL,  is  the  sine  of  ZP,  and 
therefore  the  cosine  of  the  latitude. 

39.  The  Tropics  are  the  parallels  of  latitude  that  pass  through 
the  solstices.     The  northern  tropic  is  called  the  tropic  of  Cancer ; 
the  southern,  the  tropic  of  Capricorn. 

40.  The  Polar  Circles  are  the  parallels  of  latitude  that  pass 
through  the  poles  of  the  ecliptic,  at  the  distance  of  23i  degrees 
from  the  pole  of  the  earth.     (Art.  33.) 

41.  The  earth  is  divided  into  five  zones.     That  portion  of  the 
earth  which  lies  between  the  tropics,  is  called  the  Torrid  Zone  ; 
that  between  the  tropics  and  polar  circles,  the  Temperate  Zones  ; 
and  that  between  the  polar  circles  and  the  poles,  the  Frigid 
Zones. 

42.  The  Zodiac  is  the  part  of  the  celestial  sphere  which  lies 
about  8  degrees  on  each  side  of  the  ecliptic.     This  portion  of  the 
heavens  is  thus  marked  off  by  itself,  because  the  planets  are  never 
seen  further  from  the  ecliptic  than  this  limit. 

43.  The  elevation  of  the  pole  is  equal  to  the  latitude  of  the  place. 
The  arc  PE  (Fig.  4.)=ZO.\PO=ZE  which  equals  the  lati- 
tude. 

44.  The  elevation  of  the  equator  is  equal  to  the  complement  of 
the  latitude. 

ZH=90°.    But  ZE=Lat.  /.  EH=90— Lat.=colatitude. 

45.  The  distance  of  any  place  from  the  pole  (or  the  polar  dis* 
tance)  equals  the  complement  of  the  latitude. 

EP=90°.    But  EZ=Lat. .-.  ZP=90-Lat.=colatitude. 


DIURNAL    REVOLUTION.  17 


CHAPTER    II. 

DIURNAL  REVOLUTION ARTIFICIAL   GLOBES ASTRONOMICAL 

PROBLEMS. 

46.  THE  apparent  diurnal  revolution  of  the  heavenly  bodies 
from  east  to  west,  is  owing  to  the  actual  revolution  of  the  earth 
on  its  own  axis  from  west  to  east.     If  we  conceive  of  a  radius  6T 
the  earth's  equator  extended  until  it  meets  the  concave  sphere  of 
the  heavens,  then  as  the  earth  revolves,  the  extremity  of  this  line 
would  trace  out  a  curve  on  the  face  of  the  sky,  namely,  the  celes- 
tial equator.    In  curves  parallel  to  this,  called  the  circles  of  diurnal 
revolution,  the  heavenly  bodies  actually  appear  to  move,  every  star 
having  its  own  peculiar  circle.  After  the  learner  has  first  rendered 
familiar  the  real  motions  of  the  earth  from  west  to  east,  he  may  then, 
without  danger  of  misconception,  adopt  the  common  language, 
that  all  the  heavenly  bodies  revolve  around  the  earth  once  a  day 
from  east  to  west,  in  circles  parallel  to  the  equator  and  to  each  other. 

47.  The  time  occupied  by  a  star  in  passing  from  any  point  in 
the  meridian  until  it  comes  round  to  the  same  point  again,  is  called 
a  sidereal  day,  and  measures  the  period  of  the  earth's  revolution 
on  its  axis.     If  we  watch  the  returns  of  the  same  star  from  day  to 
day,  we  shall  find  the  intervals  exactly  equal  to  one  another; 
that  is,  the  sidereal  days  are  all  equal.*     Whatever  star  we  select 
for  the  observation,  the  same  result  will  be  obtained.     The  stars, 
therefore,  always  keep  the  same  relative   position,  and   have  a 
common  movement  round  the  earth, — a  consequence  that  natu- 
rally flows  from  the  hypothesis,  that  their  apparent  motion  is  all 
produced  by  a  single  real  motion,  namely,  that  of  the  earth.     The 
sun,  moon,  and  planets,  revolve  in  like  manner,  but  their  returns  to 
the  meridian  are  not,  like  those  of  the  fixed  stars,  at  exactly  equal 
intervals. 

48.  The  appearances  of  the  diurnal  motions  of  the  heavenly 

*  Allowance  is  here  supposed  to  be  made  for  the  effects  of  precession,  &c. 

3 


18  THE    EARTH. 

bodies  are  different  in  different  parts  of  the  earth,  since  eveiy 
place  has  its  own  horizon,  (Art.  15,)  and  different  horizons  are 
variously  inclined  to  each  other.  Let  us  suppose  the  spectator 
viewing  the  diurnal  revolutions,  successively,  from  several  different 
positions  on  the  earth. 

49.  If  he  is  on  the  equator,  his  horizon  passes  through  both  poles ; 
for  the  horizon  cuts  the  celestial  vault  at  90  degrees  in  every  di- 
rection from  the  zenith  of  the  spectator ;  but  the  pole  is  likewise 
90  degrees  from  his  zenith,  and  consequently,  the  pole  must  be 
in  his  horizon.     The  celestial  equator  coincides  with  his  Prime 
Vertical,   being   a   great  circle   passing    through   the   east   and 
west  points.  Since  all  the  diurnal  circles  are  parallel  to  the  equa- 
tor, they  are  all,  like  the  equator,  perpendicular  to  his  horizon. 
Such  a  view  of  the  heavenly  bodies,  is  called  a  right  sphere ;  or, 

A  RIGHT  SPHERE  is  one  in  which  all  the  daily  revolutions  of 
the  heavenly  bodies  are  in  circles  perpendicular  to  the  horizon. 

A  right  sphere  is  seen  only  at  the  equator.  Any  star  situated 
in  the  celestial  equator,  would  appear  to  rise  directly  in  the  east,  at 
noon  to  pass  through  the  zenith  of  the  spectator,  and  to  set  directly  in 
the  west ;  in  proportion  as  stars  are  at  a  greater  distance  from  the 
equator  towards  the  pole,  they  describe  smaller  and  smaller  circles, 
until,  near  the  pole,  their  motion  is  hardly  perceptible.  In  a  right 
sphere  every  star  remains  an  equal  time  above  and  below  the  hori-- 
zon  ;  and  since  the  times  of  their  revolutions  are  equal,  the  veloci- 
ties are  as  the  lengths  of  the  circles  they  describe.  Consequently, 
as  the  stars  are  more  remote  from  the  equator  towards  the  pole, 
their  motions  become  slower,  until,  at -the  pole,  the  north  star  ap- 
pears stationary. 

50.  If  the  spectator  advances  one  degree  towards  the  north 
pole,  his  horizon  reaches  one  degree  beyond  the  pole  of  the  earth, 
and  cuts  the  starry  sphere  one  degree  below  the  pole  of  the  heav- 
ens, or  below  the  north  star,  if  that  be  taken  as  the  place  of  the 
pole.     As  he  moves  onward  towards  the  pole,  his  horizon  contin- 
ually reaches  further  and  further  beyond  it,  until  when  he  comes 
to  the  pole  of  the  earth,  and  under  the  pole  of  the  heavens,  his 
horizon  reaches  on  all  sides  to  the  equator  and  coincides  with  it. 


DIURNAL    REVOLUTION.  19 

Moreover,  since  all  the  circles  of  daily  motion  are  parallel  to  the 
equator,  they  become,  to  the  spectator  at  the  pole,  parallel  to  the 
horizon.  This  is  what  constitutes  a  parallel  sphere.  Or, 

A  PARALLEL  SPHERE  is  that  in  which  all  the  circles  of  daily 
motion  are  parallel  to  the  horizon. 

51.  To  render  this  view  of  the  heavens  familiar,  the  learner 
should  follow  round  in  his  mind  a  number  of  separate  stars,  one 
near  the  horizon,  one  a  few  degrees  above  it,  and  a  third  near  the 
zenith.    To  one  who  stood  upon  the  north  pole,  the  stars  of  the 
northern  hemisphere  would  all  be  perpetually  in  view  when  not 
obscured  by  clouds  or  lost  in  the  sun's  light,  and  none  of  those  of 
the  southern  hemisphere  would  ever  be  seen.     The  sun  would 
be  constantly  above  the  horizon  for  six  months  in  the  year,  and 
the  remaining  six  constantly  out  of  sight.     That  is,  at  the  pole 
the  days  and  nights  are  each  six  months  long.     The  phenomena 
at  the  south  pole  are  similar  to  those  at  the  north. 

52.  A  perfect  parallel  sphere  can  never  be  seen  except  at  one 
of  the  poles, — a  point  which  has  never  been  actually  reached  by 
man;  yet  the  British  discovery  ships  penetrated  within  a  few 
degrees  of  the  north  pole,  and  of  course  enjoyed  the  view  of  a 
sphere  nearly  parallel. 

53.  As  the  circles  of  daily  motion  are  parallel  to  the  horizon  of 
the  pole,  and  perpendicular  to  that  of  the  equator,  so  at  all  places 
between  the  two,  the  diurnal  motions  are  oblique  to  the  horizon. 
This  aspect  of  the  heavens  constitutes  an  oblique  sphere,  which  is 
thus  defined  : 

An  OBLIQUE  SPHERE  is  that  in  which  the  circles  of  daily  mo- 
tion are  oblique  to  the  horizon. 

Suppose  for  example  the  spectator  is  at  the  latitude  of  fifty  de- 
grees. His  horizon  reaches  50°  beyond  the  pole  of  the  earth,  and 
gives  the  same  apparent  elevation  to  the  pole  of  the  heavens.  It 
cuts  the  equator,  and  all  the  circles  of  daily  motion,  at  an  angle 
of  40°,  being  always  equal  to  the  co-altitude  of  the  pole.  Thus, 
let  HO  (Fig.  5,)  represent  the  horizon,  EQ  the  equator,  and 
PP'  the  axis  of  the  earth.  Also,  //,  mm,  &c.  parallels  of  latitude. 


20 


THE   EARTH. 


Then  the  horizon  of  a  spectator  Fig.  5. 

at  Z,  in  latitude  50°  reaches  to 
50°  beyond  the  pole  (Art.  50) ; 
and  the  angle  ECH,  is  40°.  As 
we  advance  still  further  north, 
the  elevation  of  the  diurnal  cir- 
cles grows  less  and  less,  and 
consequently  the  motions  of  the 
heavenly  bodies  more  and  more 
oblique,  until  finally,  at  the  pole, 
where  the  latitude  is  90°,  the 
angle  of  elevation  of  the  equator 
vanishes,  and  the  horizon  and  equator  coincide  with  each  other, 
as  before  stated. 

54.  The  CIRCLE  OF  PERPETUAL  APPARITION,  is  the  boundary  of 
that  space  around   the   elevated  pole,  where   the   stars   never  set. 
Its  distance  from  the  pole  is  equal  to  the  latitude  of  the  place. 
For,  since  the  altitude  of  the  pole  is  equal  to  the  latitude,  a  star 
whose  polar  distance  is  just  equal  to  the  latitude,  will  when  at  its 
lowest  point  only  just  reach  the  horizon ;  and  all  the  stars  nearer 
the  pole  than  this  will  evidently  not  descend  so  far  as  the  horizon. 

Thus,  mm  (Fig.  5,)  is  the  circle  of  perpetual  apparition,  be- 
tween which  and  the  north  pole,  the  stars  never  set,  and  its  dis- 
tance from  the  pole  OP  is  evidently  equal  to  the  elevation  of  the 
pole,  and  of  course  to  the  latitude. 

55.  In  the  opposite  hemisphere,  a  similar  part  of  the  sphere 
adjacent  to  the  depressed  pole  never  rises.     Hence, 

The  CIRCLE  OF  PERPETUAL  OCCULTATION,  is  the  boundary  of  that 
space  around  the  depressed  pole,  within  which  the  stars  never  rise. 
Thus,  m'm1  (Fig.  5,)  is  the  circle  of  perpetual  occultation,  be- 
tween which  and  the  south  pole,  the  stars  never  rise. 

56.  In  an  oblique  sphere,  the  horizon  cuts  the  circles  of  daily 
motion  unequally.     Towards  the  elevated  pole,  more  than  half 
the  circle  is  above  the  horizon,  and  a  greater  and  greater  portion 
as  the  distance  from  the  equator  is  increased,  until  finally,  within 


DIURNAL  REVOLUTION.  21 

the  circle  of  perpetual  apparition,  the  whole  circle  is  above  the 
horizon.  Just  the  opposite  takes  place  in  the  hemisphere  next 
the  depressed  pole.  Accordingly,  when  the  sun  is  in  the  equator, 
as  the  equator  and  horizon,  like  all  other  great  circles  of  the 
sphere,  bisect  each  other,  the  days  and  nights  are  equal  all  over 
the  globe.  But  when  the  sun  is  north  of  the  equator,  our  days 
become  longer  than  our  nights,  but  shorter  when  the  sun  is 
south  of  the  equator.  Moreover,  the  higher  the  latitude,  the 
greater  is  the  inequality  in  the  lengths  of  the  days  and  nights. 
All  these  points  will  be  readily  understood  by  inspecting  figure  5 

57.  Most  of  the  phenomena  of  the  diurnal  revolution  can  be 
explained,  either  on  the  supposition  that  the  celestial  sphere  actu- 
ally all  turns  around  the  earth  once  in  24  hours,  or  that  this  mo- 
tion of  the  heavens  is  merely  apparent,  arising  from  the  revolu- 
tion of  the  earth  on  its  axis  in  the  opposite  direction, — a  motion 
of  which  we  are  insensible,  as  we  sometimes  lose  the  conscious- 
ness of  our  own  motion  in  a  ship  or  a  steamboat,  and  observe  all 
external  objects  to  be  receding  from  us  with  a  common  motion. 
Proofs  entirely  conclusive  and  satisfactory,  establish  the  fact,  that 
it  is  the  earth  and  not  the  celestial  sphere  that  turns ;  but  these 
proofs  are  drawn  from  various  sources,  and  the  student  is  not  pre- 
pared to  appreciate  their  value,  or  even  to  understand  some  of 
them,  until  he  has  made  considerable  proficiency  in  the  study  of 
astronomy,  and  become  familiar  with  a  great  variety  of  astronom- 
ical phenomena.     To  such  a  period  of  our  course  of  instruction, 
we  therefore  postpone  the  discussion  of  the  hypothesis  of  the 
earth's  rotation  on  its  axis. 

58.  While  we  retain  the  same  place  on  the  earth,  the  diurnal 
revolution  occasions  no  change  in  our  horizon,  but  our  horizon 
goes  round  as  well  as  ourselves.     Let  us  first  take  our  station  on 
the  equator  at  sunrise  ;  our  horizon  now  passes  through  both  the 
poles,  and  through  the  sun,  which  we  are  to  conceive  of  as  at  a 
great  distance  from  the  earth,  and  therefore  as  cut,  not  by  the 
terrestrial  but  by  the  celestial  horizon.     As  the  earth  turns,  the 
horizon  dips  more  and  more  below  the  sun,  at  the  rate  of  15  de- 
grees for  every  hour,  and,  as  in  the  case  of  the  polar  star,  (Art.  50,) 


22  THE   EARTH. 

the  sun  appears  to  rise  at  the  same  rate.  In  six  hours,  therefore, 
it  is  depressed  90  degrees  below  the  sun,  which  brings  us  directly 
under  the  sun,  which,  for  our  present  purpose,  we  may  consider  as 
having  all  the  while  maintained  the  same  fixed  position  in  space. 
The  earth  continues  to  turn,  and  in  six  hours  more,  it  completely 
reverses  the  position  of  our  horizon,  so  that  the  western  part  of 
the  horizon  which  at  sunrise  was  diametrically  opposite  to  the 
sun  now  cuts  the  sun,  and  soon  afterwards  it  rises  above  the  level 
of  the  sun,  and  the  sun  sets.  During  the  next  twelve  hours,  the 
sun  continues  on  the  invisible  side  of  the  sphere, -until  the  hori- 
zon returns  to  the  position  from  which  it  started,  and  a  new  day 
begins. 

59.  Let  us  next  contemplate  the  similar  phenomena  at  the  poles. 
Here  the  horizon,  coinciding  as  it  does  with  the  equator,  would 
cut  the  sun  through  its  center,  and  the  sun  would  appear  to  re- 
volve along  the  surface  of  the  sea,  one  half  above  and  the  other 
half  below  the  horizon.     This   supposes   the   sun   in  its  annual 
revolution  to  be  at  one  of  the  equinoxes.     When  the  sun  is  north 
of  the   equator,  it  revolves   continually  round  in  a  path  which, 
during  a  single  revolution,  appears  parallel  to  the  equator,  and  it 
is  constantly  day ;  and  when  the  sun  is  south  of  the  equator,  it  is, 
for  the  same  reason,  continual  night. 

60.  We  have  endeavored  to  conceive  of  the  manner  in  which 
the  apparent  diurnal  movements  of  the  sun  are  really  produced  at 
two  stations,  namely,  in  the  right  sphere,  and  in  the  parallel  sphere. 
These  two  cases  being  clearly  understood,  there  will  be  little  dif- 
ficulty in  applying  a  similar  explanation  to  an  oblique  sphere 

ARTIFICIAL    GLOBES. 

61.  Artificial  globes  are  of  two  kinds,  terrestrial  and  celestial. 
The  first  exhibits  a  miniature   representation  of  the  earth ;  the 
second,  of  the  visible  heavens  ;  and  both  show  the  various  circles 
by  which  the  two  spheres  are  respectively  traversed.     Since  all 
globes  are  similar  solid  figures,  a  small  globe,  imagined  to  be  sit- 
uated at  the  center  of  the  earth  or  of  the  celestial  vault,  may  rep- 

i 


ARTIFICIAL  GLOBES.  23 

resent  all  the  visible  objects  and  artificial  divisions  of  either  sphere, 
and  with  great  accuracy  and  just  proportions,  though  on  a  scale 
greatly  reduced.  The  study  of  artificial  globes,  therefore,  cannot 
be  too  strongly  recommended  to  the  student  of  astronomy.* 

62.  An  artificial  globe  is  encompassed  from  north  to  south  by 
a  strong  brass  ring  to  represent  the  meridian  of  the  place.     This 
ring  is  made  fast  to  the  two  poles  and  thus  supports  the  globe, 
while  it  is  itself  supported  in  a  vertical  position  by  means  of  a 
frame,  the  ring  being  usually  let  into  a  socket  in  which  it  may  be 
easily  slid,  so  as  to  give  any  required  elevation  to  the  pole.     The 
brass  meridian  is  graduated  each  way  from  the  equator  to  the 
pole  90°,  to  measure  degrees  of  latitude  or  declination,  according 
as  the  distance  from  the  equator  refers  to  a  point  on  the  earth  or 
in  the  heavens.     The  horizon  is  represented  by  a  broad  zone,  made 
broad  for  the  convenience  of  carrying  on  it  a  circle  of  azimuth,  an- 
other of  amplitude,  and  a  wide  space  on  which  are  delineated  the 
signs  of  the  ecliptic,  and  the  sun's  place  for  every  day  in  the  year ; 
not  because  these  points  have  any  special  connexion  with  the  hori- 
zon, but  because  this  broad  surface  furnishes  a  convenient  place 
for  recording  them. 

63.  Hour  Circles  are  represented  on  the  terrestrial  globe  by 
great  circles  drawn  through  the  pole  of  the  equator ;  but,  on  the 
celestial  globe,  corresponding  circles  pass  through  the  poles  of  the 
ecliptic,  constituting  circles  of  celestial  latitude,  (Art.  37,)  while  the 
brass  meridian,  being  a  secondary  to  the  equinoctial,  becomes  an 
hour  circle  of  any  star  which,  by  turning  the  globe,  is  brought  un- 
der it. 

64.  The  Hour  Index  is  a  small  circle  described  around  the  pole 
of  the  equator,  on  which  are  marked  the  hours  of  the  day.     As 
this  circle  turns  along  with  the  globe,  it  makes  a  complete  revo- 
lution in  the  same  time  with  the  equator ;  or,  for  any  less  period, 

*  It  were  desirable,  indeed,  that  every  student  of  the  science  should  have  the  ce?es- 
tial  globe  at  least,  constantly  before  him.  One  of  a  small  size,  as  eight  or  nine  inches, 
will  answer  the  purpose,  although  globes  of  these  dimensions  cannot  usually  be  relied 
on  for  nice  measurements. 


24  '       THE    EARTH. 

the  same  number  of  degrees  of  this  circle  and  of  the  equator  pass 
under  the  meridian.  Hence  the  hour  index  measures  arcs  of 
right  ascension.  (Art.  37.) 

65.  The  Quadrant  of  Altitude  is  a  flexible  strip  of  brass,  gradu- 
ated into  ninety  equal  parts,  corresponding  in  length  to  degrees 
on  the  globe,  so  that  when  applied  to  the  globe  and  bent  so  as 
closely  to  fit  its  surface,  it  measures  the  angular  distance  between 
any  two  points.     When  the  zero,  or  the  point  where  the  gradua- 
tion begins,  is  laid  on  the  pole  of  any  great  circle,  the  90th  degree 
will  reach  to  the  circumference  of  that  circle,  and  being  therefore 
a  great  circle  passing  through  the  pole  of  another  great  circle,  it 
becomes  a  secondary  to  the  latter.     (Art.  21.)     Thus  the  quadrant 
of  altitude  may  be  used  as  a  secondary  to  any  great  circle  on  the 
sphere ;  but  it  is  used  chiefly  as  a  secondary  to  the  horizon,  the 
point  marked  90°  being  screwed  fast  to  the  pole  of  the  horizon, 
that  is,  the  zenith,  and  the  other  end,  marked  0,  being  slid  along 
between  the  surface  of  the  sphere  and  the  wooden  horizon.     It 
thus  becomes  a  vertical  circle,  on  which  to  measure  the  altitude 
of  any  star  through  which  it  passes,  or  from  which  to  measure 
the  azimuth  of  the  star,  which  is  the  arc  of  the  horizon  intercept- 
ed between  the  meridian  and  the  quadrant  of  altitude  passing 
through  the  star,  (Art.  27.) 

66.  To  rectify  the  globe  for  any  place,  the  north  pole  must  be 
elevated  to  the  latitude  of  the  place  (Art.  43)  ;  then  the  equator 
and  all  the  diurnal  circles  will  have  their  due  inclination  in  respect 
to  the  horizon  ;  and,  on  turning  the  globe,  (the  celestial  globe  west, 
and  the  terrestrial  east,)  every  point  on  either  globe  will  revolve  as 
the  same  point  does  in  nature ;  and  the  relative  situations  of  all 
places  will  be  the  same  as  on  the  respective  native  spheres. 

PROBLEMS  ON  THE  TERRESTRIAL  GLOBE. 

67.  To  find  the  Latitude  and  Longitude  of  a  place :  Turn  the 
globe  so  as  to  bring  the  place  to  the  brass  meridian ;  then  the  de- 
gree and  minute  on  the  meridian  directly  over  the  place  will  indi- 
cate its  latitude,  and  the  point  of  the  equator  under  the  meridian, 
will  show  its  longitude.    % 


PROBLEMS  ON  THE  TERRESTRIAL  GLOBE.  25 

Ex.  What  are  the  Latitude  and  Longitude  of  the  city  of  New 
York? 

68.  To  find  a  place  having  its  latitude  and  longitude  given:  Bring 
to  the  brass  meridian  the  point  of  the  equator  corresponding  to 
the  longitude,  and  then  at  the  degree  of  the  meridian  denoting  the 
latitude,  the  place  will  be  found. 

Ex.  What  place  on  the  globe  is  in  Latitude  39  N.  and  Longi- 
tude 77  W.  ? 

69.  To  find  the  bearing  and  distance  of  two  places:  Rectify  the 
globe  for  one  of  the  places  (Art.  66)  ;  screw  the  quadrant  of  alti- 
tude to  the  zenith,*  and  let  it  pass  through  the  other  place.     Then 
the  azimuth  will  give  the  bearing  of  the  second  place  from  the 
first,  and  the  number  of  degrees  on  the  quadrant  of  altitude,  mul- 
tiplied by  691,  (the  number  of  miles  in  a  degree,)  will  give  the 
distance  between  the  two  places. 

Ex.  What  is  the  bearing  of  New  Orleans  from  New  York,  and 
what  is  the  distance  between  these  places  ? 

70.  To  determine  the  difference  of  time  in  different  places ; 
Bring  the  place  that  lies  eastward  of  the  other  to  the  meridian, 
and  set  the  hour  index  at  XII.     Turn  the  globe  eastward  until 
the  other  place  comes  to  the  meridian,  then  the  index  will  point 
to  the  hour  required. 

Ex.  When  it  is  noon  at  New  York,  what  time  is  it  at  London  ? 

71.  The  hour  being  given  at  any  place,  to  tell  what  hour  it  is  in 
any  other  part  of  the  world :  Find  the  difference  of  time  between 
the  two  places,  (Art.  70,)  and,  if  the  place  whose  time  is  required 
is  eastward  of  the  other,  add  this  difference  to  the  given  time,  but 
if  westward,  subtract  it. 

Ex.  What  time  is  it  at  Canton,  in  China,  when  it  is  9  o'clock 
A.M.  at  New  York? 

72.  To  find  the  antceci,-\  the  periceci,  J  and  the  antipodes^  of  any 
*  The  zenith  will  of  course  be  the  point  of  the  meridian  over  the  place. 

t   aVTl  OtKOS.  |    ttfl  (HKOS.  §  CVTt  Mf. 

4 


26  THE    EARTH. 

place :  Bring  the  given  place  to  the  meridian ;  then,  under  the 
meridian,  in  the  opposite  hemisphere,  in  the  same  degree  of  lati- 
tude, will  be  found  the  antoeci.  The  same  place  remaining  under 
the  meridian,  set  the  index  to  XII,  and  turn  the  globe  until  the 
other  XII  is  under  the  index  ;  then  the  perioeci  will  be  on  the  me- 
ridian, under  the  same  degree  of  latitude  with  the  given  place, 
and  the  antipodes  will  be  under  the  meridian,  in  the  same  latitude, 
in  the  opposite  hemisphere. 

Ex.  Find  the  antoeci,  the  perioeci,  and  the  antipodes  of  the  citi- 
zens^ *New  York. 

.The  aixtG&ci  lifcve  the  same  hour  of  the  day,  but  different  seasons 
of  the  year ;  the  perioeci  have  the  same  seasons,  but  opposite  hours ; 
and  the  antipodes  have  both  opposite  hours  and  opposite  seasons. 

73.  To  ratify  the  globe  for  the  surfs  place :  On  the  wooden 
horizon,  find  the  day  of  the  month,  and  against  it  is  given  the  sun's 
place  in  the  ecliptic,  expressed  by  signs  and  degrees.*     Look  for 
the  same  sign  and  degree  on  the  ecliptic,  bring  that  point  to  the 
meridian  and  set  the  hour  index  to  XII.     To  all  places  under  the 
meridian  it  will  then  be  noon. 

Ex.  Rectify  the  globe  for  the  sun's  place  on  the  1st  of  September. 

74.  The  latitude  of  the  place  being  given,  to  find  the  time  of  the 
surfs  rising  and  setting  on  any  given  day  at  that  place :  Having 
rectified  the  globe  for  the  latitude,  (Art.  66,)  bring  the  sun's  place 
in  the  ecliptic  to  the  graduated  edge  of  the  meridian,  and  set  the 
hour  index  to  XII ;  then  turn  the  globe  so  as  to  bring  the  sun  to 
the  eastern  and  then  to  the  western  horizon,  and  the  hour  index 
will  show  the  times  of  rising  and  setting  respectively. 

Ex.  At  what  time  will  the  sun  rise  and  set  at  New  Haven, 
Lat.  41°  18',  on  the  10th  of  July  ? 

PROBLEMS  ON  THE  CELESTIAL  GLOBE. 

75.  To  find  the  Declination  and  Right  Ascension  of  a  heavenly 
body :  Bring  the  place  of  the  body  (whether  the  sun  or  a  star)  to 
the  meridian.     Then  the  degree  and  minute  standing  over  it  will 

*  The  larger  globes  have  the  day  of  the  month  marked  against  the  corresponding 
sign  on  the  ecliptic  itself. 


PROBLEMS  ON  THE  CELESTIAL  GLOBE.  27 

show  its  declination,  and  the  point  of  the  equinoctial  under  the 
meridian  will  give  its  right  ascension.  It  will  be  remarked,  that 
the  declination  and  right  ascension  are  found  in  the  same  manner 
as  latitude  and  longitude  on  the  terrestrial  globe.  Right  ascen- 
sion is  expressed  either  in  degrees  or  in  hours  ;  both  being  reck- 
oned from  the  vernal  equinox,  (Art.  37.) 

Ex.  What  is  the  declination  and  right  ascension  of  the  bright 
star  Lyra  ?  —  also  of  the  sun  on  the  5th  of  June  ? 


76.  To  represent  the  appearance  of  the  heavens  .         .^  . 
Rectify  the  globe  for  the  latitude,  bring  the  smfr  P&ce,  in 
ecliptic  to  the  meridian,  and  set  the  hour  index  to.  XII  j^hpri  ti 
the  globe  westward  until  the  index  points  to  t|i£  given  hour,  arid 
the  constellations  would  then  have  the  same  appearance  to  an  eye* 
situated  at  the  center  of  the  globe,  as  they  have  at  .  that  moment 
in  the  sky. 

Ex.  Required  the  aspect  of  the  stars  at  New  Haven,  Lat.  41° 
18',  at  10  o'clock,  on  the  evening  of  December  5th. 

77.  To  find  the  altitude  and  azimuth  of  any  star  :  Rectify  the 
globe  for  the  latitude,  and  let  the  quadrant  of  altitude  be  screwed 
to  the  zenith,  and  be  made  to  pass  through  the  star.     The  arc  on 
the  quadrant,  from  the  horizon  to  the  star,  will  denote  its  altitude, 
and  the  arc  of  the  horizon  from  the  meridian  to  the  quadrant,  will 
be  its  azimuth. 

Ex.  What  are  the  altitude  and  azimuth  of  Sinus  (the  brightest 
of  the  fixed  stars)  on  the  25th  of  December  at  10  o'clock  in  the 
evening,  in  Lat.  41°  ? 

78.  To  find  the  angular  distance  of  two  stars  from  each  other  . 
Apply  the  zero  mark  of  the  quadrant  of  altitude  to  one  of  the 
stars,  and  the  point  of  the  quadrant  which  falls  on  the  other  star, 
will  show  the  angular  distance  between  the  two. 

Ex.  What  is  the  distance  between  the  two  largest  stars  of  the 
Great  Bear?* 

*  These  two  stars  are  sometimes  called  "  the  Pointers,"  from  the  line  which  passes 
through  them  being  always  nearly  in  the  direction  of  the  north  star.  The  angular 
distance  between  them  is  about  5°,  and  may  be  learned  as  a  standard  for  reference  m 
estimating,  by  the  eye,  the  distance  between  any  two  points  on  the  celestial  vault. 


28 


PARALLAX. 


79.  To  find  the  surfs  meridian  altitude,  the  latitude  and  day 
of  the  month  being  given:  Having  rectified  the  globe  for  the 
latitude,  (Art.  66,)  bring  the  sun's  place  in  the  ecliptic  to  the  me- 
ridian, and  count  the  number  of  degrees  and  minutes  between 
that  point  of  the  meridian  and  the  zenith.  The  complement  of 
this  arc  will  be  the  sun's  meridian  altitude. 

Ex.  What  is  the  sun's  meridian  altitude  at  noon  on  the  1st  of 
August,  in  Lat.  41°  18'? 


CHAPTER   III. 

OF   PARALLAX,    REFRACTION,    AND   TWILIGHT. 

80.  PARALLAX  is  the  apparent  change  of  place  which  bodies 
undergo  by  being  viewed  from  different  points.  Thus  in  figure 
6,  let  A  represent  the  earth,  CH'  the  horizon,  H'Z  a  quadrant  of 


Fig.  6. 


a  great  circle  of  the  heavens,  extending  from  the  horizon  to  the 
zenith ;  and  let  E,  F,  G,  H,  be  successive  positions  of  the  moon 
at  different  elevations,  from  the  horizon  to  the  meridian.  Now  a 
spectator  on  the  surface  of  the  earth  at  A,  would  refer  the  place 
of  E  to  h,  whereas,  if  seen  from  the  center  of  the  earth,  it  would 


PARALLAX.  29 

appear  at  H'.  The  arc  H'h  is  called  the  parallactic  arc,  and  the 
angle  H'E£,  or  its  equal  AEC,  is  the  angle  of  -parallax.  The 
same  is  true  of  the  angles  at  F,  G,  and  H,  respectively. 

81.  Since  then  a  heavenly  body  is  liable  to  be  referred  to  dif- 
ferent points  on  the  celestial  vault,  when  seen  from  different  parts 
of  the  earth,  and  thus  some  confusion  occasioned  in  the  deter- 
mination of  points  on  the  celestial  sphere,  astronomers  have  agreed 
to  consider  the  true  place  of  a  celestial  object  to  be  that  where  it 
would  appear  if  seen  from  the  center  of  the  earth.    The  doctrine 
of  parallax  teaches  how  to  reduce  observations  made  at  any  place 
on  the  surface  of  the  earth,  to  such  as  they  would  be  if  made 
from  the  center. 

82.  The  angle  AEC  is  called  the  horizontal  parallax,  which 
may  be  thus  defined.     Horizontal  Parallax,  is  the  change  of  po- 
sition which  a  celestial  body,  appearing  in  the  horizon  as  seen 
from  the  surface  of  the  earth,  would  assume  if  viewed  from  the 
earth's  center.     It  is  the  angle  subtended  by  the  semi-diameter 
of  the  earth,  as  viewed  from  the  body  itself.     If  we  consider  any 
one  of  the  triangles  represented  in  the  figure,  ACG  for  example, 

Sin.  AGC  :  Sin.  GAZ : :  AC  :  CG 

...Sin.  ParaDa^8 

CG 

Hence  the  sine  of  the  angle  of  parallax,  or  (since  the  angle  of 
parallax  is  always  very  small*)  the  parallax  itself  varies  as  the 
sine  of  the  zenith  distance  of  the  body  directly,  and  the  distance 
of  the  body  from  the  center  of  the  earth  inversely.  Parallax,  there- 
fore, increases  as  a  body  approaches  the  horizon,  (but  increasing 
with  the  sines,  it  increases  much  slower  than  in  the  simple  ratio 
of  the  distance  from  the  zenith,)  and  diminishes,  as  the  distance 
from  the  spectator  increases.  Again,  since  the  parallax  AGC  is  as 
the  sine  of  the  zenith  distance,  let  P  represent  the  horizontal  par- 
allax, and  P'  the  parallax  at  any  altitude  ;  then, 

*  The  moon,  on  account  of  its  nearness  to  the  earth,  has  the  greatest  horizontal 
parallax  of  any  of  the  heavenly  bodies  ;  yet  this  is  less  than  1°  (being  57*)  while  the 
greatest  parallax  of  any  of  the  planets  does  not  exceed  30".  The  difference  in  an 
arc  of  1°,  between  the  length  of  the  arc  and  the  sine,  is  only  O."18. 


30  THE    EARTH. 

P 


P' :  P::sin.  zenith  dist.:  sin.  90V.P: 


sin.  zen.  dist. 

Hence,  the  horizontal  parallax  of  a  body  equals  its  parallax  at 
any  altitude,  divided  by  the  sine  of  its  distance  from  the  zenith. 

83.  From  observations,  therefore,  on  the  parallax  of  a  body  at 
any  elevation,  we  are  enabled  to  find  the  angle  subtended  by  the 
semi-diameter  of  the  earth  as  seen  from  the  body.     Or?  if  the 
horizontal  parallax  is  given,  the  parallax  at  any  altitude  may  be 
found,  for 

P'=Pxsin.  zenith  distance. 

Hence,  in  the  zenith  the  parallax  is  nothing,  and  is  at  its  max- 
imum in  the  horizon. 

84.  It  is  evident  from  the  figure,  that  the  effect  of  parallax 
upon  the  place  of  a  celestial  body  is  to  depress  it.     Thus,  in  con- 
sequence of  parallax,  E  is  depressed  by  the  arc  H'^ ;  F  by  the 
arc  P/> ;  G  by  the  arc  Rr  ;  while  H  sustains  no  change.     Hence, 
in  all  calculations  respecting  the  altitude  of  the  sun,  moon,  or  plan- 
ets, the  amount  of  parallax  is  to  be  added  ;  the  stars,  as  we  shall 
see  hereafter,  have  no  sensible  parallax.     As  the  depression  which 
arises  from  parallax  is  in  the  direction  of  a  vertical  circle,  a  body, 
when  on  the  meridian,  has  only  a  parallax  in  declination ;   but 
in  other   situations,   there   is   at  the   same   time  a  parallax  in 
declination  and  right  ascension  ;  for  its  direction  being  oblique 
to  the  equinoctial,  it  can  be  resolved  into  two  parts,  one  of  which 
(the  declination)  is  perpendicular,  and  the  other  (the  right  ascen- 
sion) is  parallel  to  the  equinoctial. 

85.  The   mode  of  determining   the  horizontal  parallax,   is  as 
follows : 

Let  O,  O',  (Fig.  7,)  be  two  places  on  the  earth,  situated  under 
the  same  meridian,  at  a  great  distance  from  each  other  ;  one  place, 
for  example,  at  the  Cape  of  Good  Hope,  and  the  other  in  the  north 
of  Europe.  The  latitude  of  each  place  being  known,  the  arc  of 
the  meridian  OO'  is  known,  and  the  angle  OCO'  also  is  known. 
Let  the  celestial  body  M,  (the  moon  for  example,)  he  observed 
simultaneously  at  O  and  O',  and  its  zenith  distance  at  each  place 


PARALLAX. 


31 


accurately  taken,  namely,  ZY  and 

Z'Y'  ;  then  the  angles  ZOM  and 

Z'O'M,  and  of  course  their  sup- 

plements COM,CO'M  are  found. 

Then  in  the  quadrilateral  figure 

COMO',  we  have  all  the  angles 

and    the    two   radii,  CO,  CO7, 

whence  by  joining  OO',  the  side 

OM  may  be  easily  found.   Hav- 

ing CO  and  OM,  we  may  find 

CMO=sine  of  the  angle  of  par- 

allax ;  or  (since  the  arc  is  very 

small)   equals    the  parallax    P'. 

But  when  M  as  seen  from  O  is  in  the  horizon,  ZOM  becomes  a 

right  angle,  and  its  sine  equal  to  radius.     Then,  CM  being  found, 


CM  :  CO  :  :  1  :  P=horizontal  parallax=. 

CM 

On  this  principle,  the  horizontal  parallax  of  the  moon  was  de- 
termined by  La  Caille  and  La  Lande,  two  French  astronomers, 
one  stationed  at  the  Cape  of  Good  Hope,  the  other  at  Berlin  ;  and 
in  a  similar  way  the  parallax  of  Mars  was  ascertained,  by  ob- 
servations made  simultaneously  at  the  Cape  of  Good  Hope  and 
at  Stockholm. 

86.  On  account  of  the  great  distance  of  the  sun,  his  horizontal 
parallax,  which  is  only  8".6,  cannot  be  accurately  ascertained  by 
this  method.     It  can,  however,  be  determined  by  means  of  the 
transits  of  Venus,  a  process  to  be  described  hereafter. 

87.  The  determination  of  the  horizontal  parallax  of  a  celestial 
body  is  an  element  of  great  importance,  since  it  furnishes  the 
means  of  estimating  the  distance  of  the  body  from  the  center  of 
the  earth.     Thus,  if  the  angle  AEC  (Fig.  6,)  be  found,  the  radius 
of  the  earth  AC  being  known,  we  have  in  the  triangle  AEC, 
right  angled  at  A,  the  side  AC  and  all  the  angles,  to  find  the  hypo- 
thenuse  CE,  which  is  the  distance  of  the  moon  from  'he  center 
of  the  earth. 


32  THE    EARTH. 

REFRACTION. 

88.  While  parallax  depresses  the  celestial  bodies  subject  to  it, 
refraction  elevates  them;  and  it  affects  alike  the  most  distant 
as  well  as  nearer  bodies,  being  occasioned  by  the  change  of  di- 
rection which  light  undergoes  in  passing  through  the  atmos- 
phere. Let  us  conceive  of  the  atmosphere  as  made  up  of  a  great 
number  of  concentric  strata,  as  AA,  BB,  CC,  and  DD,  (Fig.  8,) 

Fig.  8. 


increasing  rapidly  in  density  (as  is  known  to  be  the  fact)  in  ap- 
proaching near  to  the  surface  of  the  earth.  Let  S  be  a  star,  from 
which  a  ray  of  light  S«  enters  the  atmosphere  at  #,  where,  being 
turned  towards  the  radius  of  the  convex  surface,  it  would  change 
its  direction  into  the  line  ab,  and  again  into  be,  and  cO,  reach- 
ing the  eye  at  O.  Now,  since  an  object  always  appears  in  the 
direction  in  which  the  light  finally  strikes  the  eye,  the  star  would 
be  seen  in  the  direction  of  the  last  ray  cO,  and  the  star  would 
apparently  change  its  place,  in  consequence  of  refraction,  from 
S  to  S',  being  elevated  out  of  its  true  position.  Moreover, 
since  on  account  of  the  constant  increase  of  density  in  descend- 
ing through  the  atmosphere,  the  light  would  be  continually  turned 
out  of  its  course  more  and  more,  it  would  therefore  move,  not 
in  the  polygon  represented  in  the  figure,  but  in  a  corresponding 
curve,  whose  curvature  is  rapidly  increased  near  the  surface  of 
the  earth. 

89.  When  a  body  is  in  the  zenith,  since  a  ray  of  light  from  it 
enters  the  atmosphere  at  right  angles  to  the  refracting  medium,  it 
suffers  no  refraction.  Consequently,  the  position  of  the  heavenly 


REFRACTION. 


bodies,  when  in  the  zenith,  is  not  changed  by  refraction,  while, 
near  the  horizon,  where  a  ray  of  light  strikes  the  medium  very 
obliquely,  and  traverses  the  atmosphere  through  its  densest  part, 
the  refraction  is  greatest.  The  following  numbers,  taken  at  dif- 
ferent altitudes,  will  show  how  rapidly  refraction  diminishes  from 
the  horizon  upwards.  The  amount  of  refraction  at  the  horizon 
is  34-  00".  At  different  elevations  it  is  as  follows. 


Elevation. 

Refraction. 

Elevation. 

Refraction. 

0°   10' 

32'  00" 

30° 

1'  40" 

0    20 

30    00 

40 

1    09 

1    00 

24    25 

45 

0    58 

5    00 

10    00 

60 

0    33 

10    00 

5    20 

80 

0    10 

20    00 

2    39 

90 

0    00 

From  this  table  it  appears,  that  while  refraction  at  the  horizon 
is  34  minutes,  at  so  small  an  elevation  as  only  10  minutes  above 
the  horizon  it  loses  2  minutes,  more  than  the  entire  change  from 
the  elevation  of  30°  to  the  zenith.  From  the  horizon  to  1°  above, 
the  refraction  is  diminished  nearly  10  minutes.  The  amount  at 
the  horizon,  at  45°,  and  at  90°,  respectively,  is  34',  58",  and  0.  In 
finding  the  altitude  of  a  heavenly  body,  the  effect  of  parallax  must 
be  added,  but  that  of  refraction  subtracted. 

90.  Let  us  now  learn  the  method,  by  which  the  amount  of  re- 
fraction at  different  elevations  is  ascertained.  To  take  the  sim- 
plest case,  we  will  suppose  ourselves  in  a  high  latitude,  where 
some  of  the  stars  within  the  circle  of  perpetual  apparition  pass 
through  the  zenith  of  the  place.  We  measure  the  distance  of 
such  a  star  from  the  pole  when  on  the  meridian  above  the  pole, 
that  is,  in  the  zenith,  where  it  is  not  at  all  affected  by  refraction, 
and  again  its  distance  from  the  pole  in  its  lower  culmination. 
Were  it  not  for  refraction,  these  two  polar  distances  would  be 
equal,  since,  in  the  diurnal  revolution  of  a  star,  it  is  in  fact  always 
at  the  same  distance  from  the  pole ;  but,  on  account  of  refraction, 
the  lower  distance  will  be  less  than  the  upper,  and  the  difference 
between  the  two  will  show  the  amount  of  refraction  at  the  lower 
culmination,  the  latitude  of  the  place  being  known. 

Example.     At  Paris,  latitude  48°  50',  a  star  was  observed  to 
5 


34 


THE    EARTH. 


pass  the  meridian  &  north  of  the  zenith,  and  consequently,  41°  4' 
from  the  pole.*  It  should  have  passed  the  meridian  at  the  same 
distance  below  the  pole,  but  the  distance  was  found  to  be  only 
40°  57'  35".  Hence,  41°  4'-40°  57'  35"=6'  25"  is  the  refraction 
due  to  that  altitude,  that  is,  at  the  altitude  of  7°  46'=(48°  50'- 
41°  4').  By  taking  similar  observations  in  various  places  situated 
in  high  latitudes,  the  amount  of  refraction  might  be  ascertained 
for  a  number  of  different  altitudes,  and  thus  the  law  of  increase 
in  refraction  as  we  proceed  from  the  zenith  towards  the  horizon, 
might  be  ascertained. 

91.  Another  method  of  finding  the  refraction  at  different  alti- 
tudes, is  as  follows.  Take  the  altitude  of  the  sun  or  a  star,  whose 
right  ascension  and  declination  are  known,  and  note  the  time  by 
the  clock.  Observe  also  when  it  crosses  the  meridian,  and  the 
difference  of  time  between  the  two  observations  will  give  the  hour 
angle  ZPx,  (Fig.  9.)  In  this  triangle  ZPx  we  also  know  PZ  the 

Fig.  9. 


co-latitude  and  Pa?  the  co-declination.     Hence  we  can  find  the  co 
altitude  Zx,  and  of  course  the  true  altitude.     Compare  the  alti- 
tude thus  found  with  that  before  determined  by  observation,  and 
the  difference  will  be  the  refraction  due  to  the  apparent  altitude. 

*For  the  polar  distance  of  the  place=90-48°  50'=4P  10';  and  41<>  10'-6'= 


REFRACTION.  35 

Ex.  On  May  1,  1738,  at  5h.  20m.  in  the  morning,  Cussini  ob- 
served the  altitude  of  the  sun's  center  at  Paris  to  be  5°  0'  14".  The 
latitude  of  Paris  being  48°  50'  10",  and  the  sun's  declination  at 
that  time  being  15°  0'  25"  :  Required  the  refraction. 

By  spherical  trigonometry,  Zx  will  be  found  equal  to  85°  10' 
8";  consequently,  the  true  altitude  was  4°  49'  52".  Now  to  5° 
0'  14",  the  apparent  altitude,  9"  must  be  added  for  parallax, 
and  we  have  5°  0'  23"  the  apparent  altitude  corrected  for  parallax. 
Hence,  5°  0'  23"-4°  49'  52"=10'  31",  the  refraction  at  the  ap- 
parent altitude  5°  0'  14".* 

92.  By  these  and  similar  methods,  we  could  easily  determine 
the  refraction  due  to  any  elevation  above  the  horizon,  provided 
the  refracting  medium  (the  atmosphere)  were  always  uniform. 
But  this  is  not  the  fact :  the  refracting  power  of  the  atmosphere 
is  altered  by  changes  in  density  and  temperature. f     Hence  in 
delicate  observations,  it  is  necessary  to  take  into  the  account  the 
state  of  the  barometer  and  of  the  thermometer,  the  influence  of 
the  variations  of  each  having  been  very  carefully  investigated, 
and  rules  having  been  given  accordingly.     With  every  precaution 
to  insure  accuracy,  on  account  of  the  variable  character  of  the 
refracting  medium,  the  tables  are  not  considered  as  entirely  accu- 
rate to  a  greater  distance  from  the  zenith  than  74°  ;  but  almost  all 
astronomical  observations  are  made  at  a  greater  altitude  than  this. 

93.  Since  the  whole  amount  of  refraction  near  the  horizon  ex- 
ceeds 33',  and  the  diameters  of  the  sun  and  moon  are  severally 
less  than  this,  these  luminaries  are  in  view  both  before  they  have 
actually  risen  and  after  they  have  set. 

94.  The  rapid  increase  of  refraction  near  the  horizon,  is  strik- 
ingly evinced  by  the  oval  figure  which  the  sun  assumes  when 
near  the  horizon,  and  which  is  seen  to  the  greatest  advantage 
when  light  clouds  enable  us  to  view  the  solar  disk.     Were  all , 

*  Gregory's  Ast.  p.  65. 

t  It  is  said  that  the  effects  of  humidity  are  insensible ;  for  the  most  accurate 
experiments  seem  to  prove  that  watery  vapor  diminishes  the  density  of  air  in  the 
same  ratio  as  its  own  refractive  power  is  greater  than  that-  of  air.  (New  Encyc. 
Brit.  Ill,  762.) 


36  THE    EARTH. 

parts  of  the  sun  equally  raised  by  refraction,  there  would  be  no 
change  of  figure ;  but  since  the  lower  side  is  more  refracted  than 
the  upper,  the  effect  is  to  shorten  the  vertical  diameter  and  thus 
to  give  the  disk  an  oval  form.  This  effect  is  particularly  remark- 
able when  the  sun,  at  his  rising  or  setting,  is  observed  from  the 
top  of  a  mountain,  or  at  an  elevation  near  the  sea  shore  ;  for  in 
such  situations  the  rays  of  light  make  a  greater  angle  than  or- 
dinary with  a  perpendicular  to  the  refracting  medium,  and  the 
amount  of  refraction  is  proportionally  greater.  In  some  cases  of 
this  kind,  the  shortening  of  the  vertical  diameter  of  the  sun  has 
been  observed  to  amount  to  6',  or  about  one  fifth  of  the  whole.* 

95.  The  apparent  enlargement  of  the  sun  and  moon  in  the  hori- 
zon, arises  from  an  optical  illusion.  These  bodies  in  fact  are 
not  seen  under  so  great  an  angle  when  in  the  horizon,  as  when  on 
the  meridian,  for  they  are  nearer  to  us  in  the  latter  case  than  in 
the  former.  The  distance  of  the  sun  is  indeed  so  great  that  it 
makes  very  little  difference  in  his  apparent  diameter,  whether  he 
is  viewed  in  the  horizon  or  on  the  meridian ;  but  with  the  moon 
the  case  is  otherwise ;  its  angular  diameter,  when  measured  with 
instruments,  is  perceptibly  larger  at  the  time  of  its  culmination. 
Why  then  do  the  sun  and  moon  appear  so  much  larger  when  near 
the  horizon  ?  It  is  owing  to  that  general  law,  explained  in  optics, 
by  which  we  judge  of  the  magnitudes  of  distant  objects,  not 
merely  by  the  angle  they  subtend  at  the  eye,  but  also  by  our  im- 
pressions respecting  their  distance,  allowing,  under  a  given  angle, 
a  greater  magnitude  as  we  imagine  the  distance  of  a  body  to  be 
greater.  Now,  on  account  of  the  numerous  objects  usually  in 
sight  between  us  and  the  sun,  when  on  the  horizon,  he  appears 
much  further  removed  from  us  than  when  on  the  meridian,  and 
we  assign  to  him  a  proportionally  greater  magnitude.  If  we  view 
the  sun,  in  the  two  positions,  through  smoked  glass,  no  such  dif- 
ference of  size  is  observed,  for  here  no  objects  are  seen  but  the 
sun  himself. 


*  In  extreme  cold  weather,  this  shortening  of  the  sun's  vertical  diameter  sometimes 
exceeds  this  amount. 


TWILIGHT.  37 

TWILIGHT. 

96.  Twilight  also  is  another  phenomenon  depending  upon  the 
agency  of  the  earth's  atmosphere.  It  is  due  partly  to  refraction 
and  partly  to  reflexion,  but  mostly  the  latter.  While  the  sun 
is  within  18°  of  the  horizon,  before  it  rises  or  after  it  sets,  some 
portion  of  its  light  is  conveyed  to  us  by  means  of  numerous  re- 
flections from  the  atmosphere.  Let  AB  (Fig.  10,)  be  the  horizon 

Fig.  10. 


of  the  spectator  at  A,  and  let  SS  be  a  ray  of  light  from  the  sun 
when  it  is.  two  or  three  degrees  below  the  horizon.  Then  to 
the  observer  at  A,  the  segment  of  the  atmosphere  ABS  would  be 
illuminated.  To  a  spectator  at  C,  whose  horizon  was  CD,  the 
small  segment  SDx  would  be  the  twilight ;  while,  at  E,  the  twi- 
light would  disappear  altogether. 

97.  At  the  equator,  where  the  circles  of  daily  motion  are  per- 
pendicular to  the  horizon,  the  sun  descends  through  18°  in  an 
hour  and  twelve  minutes  (}f =l}h.),  and  the  light  of  day  there- 
fore declines  rapidly,  and  as  rapidly  advances  after  daybreak  in  the 
morning.     At  the  pole,  a  constant  twilight  is  enjoyed  while  the  sun 
is  within  18°  of  the  horizon,  occupying  nearly  two  thirds  of  the 
half  year  when  the  direct  light  of  the  sun  is  withdrawn,  so  that 
the  progress  from  continual  day  to  constant  night  is  exceedingly 
gradual.     To  the  inhabitants  of  an  oblique  sphere,  the  twilight 
is  longer  in  proportion  as  the  place  is  nearer  the  elevated  pole. 

98.  Were  it  not  for  the  power  the  atmosphere  has  of  dispersing 


38  THE   EARTH. 

the  solar  light,  and  scattering  it  in  various  directions,  no  objects 
would  be  visible  to  us  out  of  direct  sunshine  ;  every  shadow  of  a 
passing  cloud  would  be  pitchy  darkness  ;  the  stars  would  be  visi- 
ble all  day,  and  every  apartment  into  which  the  sun  had  not  di- 
rect admission,  would  be  involved  in  the  obscurity  of  night.  This 
scattering  action  of  the  atmosphere  on  the  solar  light,  is  greatly 
increased  by  the  irregularity  of  temperature  caused  by  the  sun, 
which  throws  the  atmosphere  into  a  constant  state  of  undulation, 
and  by  thus  bringing  together  masses  of  air  of  different  tempera- 
tures, produces  partial  reflections  and  refractions  at  their  common 
boundaries,  by  which  means  much  light  is  turned  aside  from  the 
direct  course,  and  diverted  to  the  purposes  of  general  illumination.* 
In  the  upper  regions  of  the  atmosphere,  as  on  the  tops  of  very 
high  mountains,  where  the  air  is  too  much  rarefied  to  reflect  much 
light,  the  sky  assumes  a  black  appearance,  and  stars  become  visi- 
ble in  the  day  time. 


CHAPTER  IV. 

OF  TIME. 

99.  TIME  is  a  measured  portion  of  indefinite  duration. 

Any  event  may  be  taken  as  a  measure  of  time,  which  divides 
a  portion  of  duration  into  equal  parts ;  as  the  pulsations  of  the 
wrist,  the  vibrations  of  a  pendulum,  or  the  passage  of  sand  from 
one  vessel  into  another,  as  in  the  hour-glass. 

100.  The  great  standard  of  time  is  the  period  of  the  revolution 
of  the  earth  on  its  axis,  which,  by  the  most  exact  observations,  is 
found  to  be  always  the  same.     The  time  of  the  earth's  revolution 
on  its  axis  is  called  a  sidereal  day,  and  is  determined  by  the  revo- 
lution of  a  star  from  the  instant  it  crosses  the  meridian,  until  it 
comes  round  to  the  meridian  again.     This  interval  being  called  a 

*  Herschel. 


TIME.  39 

sidereal  day*  it  is  divided  into  24  sidereal  hours.  Observations 
taken  upon  numerous  stars,  in  different  ages  of  the  world,  show 
that  they  all  perform  their  diurnal  revolutions  in  the  same  time, 
and  that  their  motion  during  any  part  of  the  revolution  is  per- 
fectly uniform. 

101.  Solar  time  is  reckoned  by  the  apparent  revolution  of  the 
sun,  from  the  mefldian  round  to  the  same  meridian  again.     Were 
the  sun  stationary  in  the  heavens,  like  a  fixed  star,  the  time  of  its 
apparent  revolution  would  be  equal  to  the  revolution  of  the  earth 
on  its  axis,  and  the  solar  and  the  sidereal  days  would  be  equal. 
But  since  the  sun  passes  from  west  to  east,  through  360°  in  365^ 
days,  it  moves   eastward  nearly   1°  a  day,  (59'  8". 3).     While, 
therefore,  the  earth  is  turning  round  on  its  axis,  the  sun  is  moving 
in  the  same  direction,  so  that  when  we  have  come  round  under 
the  same  celestial  meridian  from  which  we  started,  we  do  not 
find  the  sun  there,  but  he  has  moved  eastward  nearly  a  degree, 
and  the  earth  must  perform  so  much  more  than  one  complete 
revolution,  in  order  to  come  under  the  sun  again.     Now  since  a 
place  on  the  earth  gains  359°  in  24  hours,  how  long  will  it  take 
to  gain  1°  1 

94 

359 :  24 : :  1 :  — =4m  nearly. 

Hence  the  solar  day  is  about  4  minutes  longer  than  the  sidereal; 
and  if  we  were  to  reckon  the  sidereal  day  24  hours,  we  should 
reckon  the  solar  day  24h.  4m.  To  suit  the  purposes  of  society  at 
large,  however,  it  is  found  most  convenient  to  reckon  the  solar  day 
24  hours,  and  to  throw  the  fraction  into  the  sidereal  day.  Then, 

24h.  4m.  :  24  :  :  24 :  23h.  56m.  (23h.  56m  4S.09)  =  the  length 
of  a  sidereal  day. 

102.  The  solar  days,  however,  do  not  always  differ  from  the 
sidereal  by  precisely  the  same  fraction,  since  the  increments  of 
right  ascension,  (Art.  37,)  which  measure  the  difference  between 
a  sidereal  and  a  solar  day,  are  not  equal  to  each  other.     Apparent 
time,  is  time  reckoned  by  the  revolutions  of  the  sun  from  tne 
meridian  to  the  meridian  again.     These  intervals  being  unequal 
of  course  the  apparent  solar  days  are  unequal  to  each  other. 


40  THE   EARTH. 

103.  Mean  time,  is  time  reckoned  by  the  average  length  of  all 
the  solar  days  throughout  the  year.     This  is  the  period  which  con- 
stitutes the  civil  day  of  24  hours,  beginning  when  the  sun  is  on 
the  lower  meridian,  namely,  at  12  o'clock  at  night,  and  counted 
by  12  hours  from  the  lower  to  the  upper  culmination,  and  from 
the  upper  to  the  lower.     The  astronomical  day  is  the  apparent 
solar  day  counted  through  the  whole  24  hours,  Listead  of  by  pe- 
riods of  12  hours  each,  and  begins  at  noon.     Tnus  10  days  and 
14  hours  of  astronomical  time,  would  be  1 1  days  and  2  hours  of 
civil  time. 

104.  Clocks  are  usually  regulated  so  as  to  indicate  mean  solar 
time  ;  yet  as  this  is  an  artificial  period,  not  marked  off,  like  the 
sidereal  day,  by  any  natural  event,  it  is  necessary  to  know  how 
much  is  to  be  added  to  or  subtracted  from  the  apparent  solar 
time,  in  order  to  give  the  corresponding  mean  time.     The  inter- 
val by  which  apparent  time  diners  from  mean  time,  is  called  the 
equation  of  time.     If  a  clock  were  constructed  (as  it  may  be)  so 
as  to  keep  exactly  with  the  sun,  going  faster  or  slower  according 
as  the  increments  of  right  ascension  were  greater  or  smaller,  and 
another  clock  were  regulated  to  mean  time,  then  the  difference 
of  the  two  clocks,  at  any  period,  would  be  the  equation  of  time 
for  that  moment.     If  the  apparent  clock  were  faster  than  the 
mean,  then  the  equation  of  time  must  be  subtracted  ;  but  if  the 
apparent  clock  were  slower  than  the  mean,  then  the  equation  of 
time  must  be  added,  to  give  the  mean  time.     The  two  clocks 
would  differ  most  about  the  3d  of  November,  when  the  apparent 
time  is  16£m  greater  than  the  mean  (16m  178).     But,  since  appa- 
rent time  is  sometimes  greater  and  sometimes  less  than  mean 
time,  the  two  must  obviously  be  sometimes  equal  to  each  other. 
This  is  in  fact  the  case  four  times  a  year,  namely,  April  15th, 
June   15th,  September  1st,  and  December  22d.     These  epochs, 
however,  do  not  remain  constant ;  for,  on  account  of  the  change 
in  the  position  of  the  perihelion,  or  the  point  where  the  earth  is 
nearest  the  sun,  (which  shifts  its  place  from  west  to  east  about 
12"  a  year,)  the  period  when  the  sun's  motions  are  most  rapid,  as 
well  as  that  when  they  are  slowest,  will  occur  at  different  parts  of 
the  year.     The  change  is  indeed  exceedingly  small  in  a  single 


TIME. 


41 


year ;  but  in  the  progress  of  ages,  the  time  of  year  when  the  sun's 
motion  in  its  orbit  is  most  accelerated,  will  not  be,  as  at  present,  on 
the  first  of  January,  but  may  fall  on  the  first  of  March,  June,  or 
any  other  day  of  the  year,  and  the  amount  of  the  equation  of 
time  is  obviously  affected  by  the  sun's  distance  from  its  perihelion, 
since  the  sun  moves  most  rapidly  when  nearest  the  perihelion,  and 
slowest  when  furthest  from  that  point. 

105.  The  inequality  of  the  solar  days  depends  on  two  causes,  the 
unequal  motion  of  the  earth  in  its  orbit,  and  the  inclination  of  the 
equator  to  the  ecliptic. 

First,  on  account  of  the  eccentricity*  of  the  earth's  orbit,  the 
earth  actually  moves  faster  from  the  autumnal  to  the  vernal  equi- 
nox, than  from  the  vernal  to  the  autumnal,  the  difference  of  the 
two  periods  being  about  eight  days  (7d.  17h.  17m.)  Thus,  let 

Fig.  11. 


AEB  (Fig.  11,)  represent  the  earth's  orbit,  S  being  the  place  ol 


*  The  exact  figure  of  the  earth's  orbit  will  be  more  particularly  shown  hereafter. 
All  that  the  student  requires  to  know,  in  order  to  understand  the  present  subject, 

6 


42  THE    EARTH. 

the  sun,  A  the  perihelion,  or  nearest  distance  of  the  earth  from 
the  sun,  B  the  aphelion,  or  greatest  distance,  and  E,  E',  E'',  posi- 
tions of  the  earth  in  different  points  of  its  orbit.  The  place  of 
the  earth  among  the  signs  is  the  part  of  the  heavens  to  which  it 
would  be  referred  if  seen  from  the  sun ;  and  the  place  of  the  sun 
is  the  part  of  the  heavens  to  which  it  is  referred  as  seen  from  the 
earth.  Thus,  when  the  earth  is  at  E,  it  is  said  to  be  in  Aries ; 
and  as  it  moves  from  E  through  E'  to  A,  its  path  in  the  heavens 
is  through  Aries,  Taurus,  Gemini,  &c.  Meanwhile  the  sun  takes 
its  place  successively  in  Libra,  Scorpio,  Sagittarius,  &c.  Now, 
as  will  be  shown  more  fully  hereafter,  the  earth  moves  faster 
when  proceeding  from  Aries  through  its  perihelion  to  Libra,  than 
from  Libra  through  its  aphelion  to  Aries,  and,  consequently,  de- 
scribes the  half  of  its  apparent  orbit  in  the  heavens,  T,  55,  =£*, 
sooner  than  the  half  =£=,  V3,  T.  The  line  of  the  apsides,  that  is, 
the  major  axis  of  the  ellipse,  is  so  situated  at  present,  that  the 
perihelion  is  in  the  sign  Cancer,  nearly  100°  (99°  30'  5")  from  the 
vernal  equinox.  The  earth  passes  through  it  about  the  first  of 
January,  and  then  its  velocity  is  the  greatest  in  the  whole  year, 
being  always  greater  as  the  distance  is  less,  the  angular  velocity 
being  inversely  as  the  square  of  the  distance,  as  will  be  shown  by 
and  by. 

106.  But  differences  of  time  are  not  reckoned  on  the  eclip- 
tic, but  on  the  equinoctial ;  for  the  ecliptic  being  oblique  to  the 
meridian  in  the  diurnal  motion,  and  cutting  it  at  different  angles  at 
different  times,  equal  portions  will  not  pass  under  the  meridian  in 
equal  times,  and  therefore  such  portions  could  not  be  employed,  as 
they  are  in  the  equinoctial,  as  measures  of  time.  If  therefore  the 
sun  moved  uniformly  in  his  orbit,  so  as  to  make  the  daily  incre- 
ments of  longitude  equal,  still  the  corresponding  arcs  of  right  as- 
cension, which  determine  the  lengths  of  the  solar  day,  would  be 
unequal.  Let  us  start  from  the  equinox,  from  which  both  longi- 
tude and  right  ascension  are  reckoned,  the  former  on  the  ecliptic, 


is  that  the  earth's  orbit  is  an  ellipse,  and  that  the  earth's  real  motion,  and  conse- 
quently the  sun's  apparent  motion,  is  greater  in  proportion  as  the  earth  is  nearer 
the  sun. 


TIME. 


43 


the  latter  on  the  equinoctial.  Suppose  the  sun  has  described  70° 
of  longitude  ;  then  to  ascertain  tne  corresoonding  arc  of  right  as- 
cension, we  let  a  meridian  pass  through  tne  sun  :  the  point  where 
it  cuts  the  equator  gives  the  sun's  right  ascension.  Now  since  the 
ecliptic  makes  an  acute  angle  with  the  meridian,  while  the  equi- 
noctial makes  a  right  angle  with  it,  consequently  the  arc  of  longi- 
tude is  greater  than  the  arc  of  right  ascension.  The  difference, 
however,  grows  constantly  less  and  less  as  we  approach  the  tropic, 
as  the  angle  made  between  the  ecliptic  and  the  meridian  constantly 
increases,  until,  when  we  reach  the  tropic,  the  meridian  is  at  right 
angles  to  both  circles,  and  the  longitude  and  right  ascension  each 
equals  90°,  and  they  are  of  course  equal  to  each  other.  Beyond 
this,  from  the  tropic  to  the  other  equinox,  the  arc  of  the  ecliptic 
intercepted  between  the  meridian  and  the  autumnal  equinox  being 
greater  than  the  corresponding  arc  of  the  equinoctial,  of  course 
its  supplement,  which  measures  the  longitude,  is  less  than  the  sup- 
plement of  the  corresponding  arc  of  the  equator  which  measures 
the  right  ascension.  At  the  autumnal  equinox  again,  the  right 
ascension  and  longitude  become  equal.  In  a  similar  manner  we 
might  show  that  the  daily  increments  of  longitude  and  right  as- 
cension are  unequal. 

In  order  to  illustrate  the  foregoing  points,  let  T  ***  (Fig.  12,) 

Fig.  12. 


represent  the  equator,  T  T  =*=  the  ecliptic,  and  PSE,  PS'E',  two 
meridians  meeting  the  sun  in  S  and  S'.   Then  in  the  triangle  TES, 


44  THE   EARTH. 

the  arc  of  longitude  TS,  is  greater  than  TE,  the  corresponding 
arc  of  right  ascension;  but  towards  the  tropic  the  difference 
between  the  two  arcs  evidently  grows  less  and  less,  until  at  T 
the  arcs  become  equal,  being  each  90°.  But,  beyond  the  tropic, 
since  TE'===,  TS'^=,  are  equal  to  each  other,  each  being  equal 
to  180°,  and  since  S'=^=  is  greater  than  E'=s=,  therefore  TS'  must 
be  less  than  TE'. 

107.  As  the  whole  arc  of  right  ascension  reckoned  from  the 
first  of  Aries,  does  not  keep  uniform  pace  with  the  corresponding 
arc  of  longitude,  so  the  daily  increments  of  right  ascension  differ 
from  those  of  longitude.     If  we  suppose  in  the  quadrant  TT, 
points  taken  to  mark  the  progress  of  the  sun  from  day  to  day,  and 
let  meridians  like  PSE  pass  through  these  points,  the  arc  of  the 
ecliptic  embraced  between  the  meridians  will  be  the  daily  incre- 
ments of  longitude,  while  the  corresponding  parts  of  the  equinoc- 
tial will  be  the  daily  increments  of  right  ascension.     Near  T,  the 
oblique  direction  in  which  the  ecliptic  cuts  the  meridian,  will  make 
the  daily  increments  of  longitude  exceed  those  of  right  ascension ; 
but  this  advantage  is  diminished  as  we  approach  the  tropic,  where 
the  ecliptic  becomes  less  oblique,  and  finally  parallel  to  the  equi- 
noctial ;  while  the  convergence  of  the  meridians  contributes  still 
farther  to  lessen  the  ratios  of  the  daily  increments  of  longitude  to 
those  of  right  ascension.     Hence,  at  first,  the  diurnal  arcs  of 
right  ascension  are  less  than  those  of  longitude,  but  afterwards 
greater ;  and  they  continue  greater  for  a  similar  distance  beyond 
the  tropic. 

108.  From  the   foregoing  considerations  it  appears,  that  the 
diurnal  arcs  of  right  ascension,  by  which  the  difference  between 
the  sidereal  and  the  solar  days  is  measured,  are  unequal,  on  ac- 
count both  of  the  unequal  motion  of  the  sun  in  his  orbit,  and  of 
the  inclination  of  his  orbit  to  the  equinoctial. 

109.  As  astronomical  time  commences  when  the  sun  is  on  the 
meridian,  so  sidereal  time  commences  when  the  vernal  equinox 
is  on  the  meridian,  and  is  also  counted  from  0  to  24  hours.     By 
3  o'clock,  for  instance,  of  sidereal  time,  we  mean  that  it  is  three 


THE    CALENDAR.  45 

hours  since  the  vernal  equinox  crossed  the  meridian ;  as  we  say  it 
is  3  o'clock  of  astronomical  or  of  civil  time,  when  it  is  three  hours 
since  the  sun  crossed  the  meridian. 


THE   CALENDAR. 

110.  The  astronomical  year  is  the  time  in  which  the  sun  makes 
one  revolution  in  the  ecliptic,  and  consists  of  365d.  5h.  48m.  5P.60. 
The  civil  year  consists  of  365  days.     The  difference  is  nearly  6 
hours,  making  one  day  in  four  years. 

111.  The  most  ancient  nations  determined  the  number  of  days 
in  the  year  by  means  of  the  stylus,  a  perpendicular  rod  which 
cast  its  shadow  on  a  smooth  plane,  bearing  a  meridian  line.     The 
time  when  the  shadow  was  shortest,  would  indicate  the  day  of 
the  summer  solstice ;  and  the  number  of  days  which  elapsed  until 
the  shadow  returned  to  the  same  length  again,  would  show  the 
number  of  days  in  the  year.     This  was  found  to  be  365  whole 
days,  and  accordingly  this  period  was  adopted  for*the  civil  year. 
Such  a  difference,  however,  between  the  civil  and  astronomical 
years,  at  length  threw  all  dates  into  confusion.     For,  if  at  first 
the  summer  solstice  happened  on  the  21st  of  June,  at  the  end  of 
four  years,  the  sun  would  not  have  reached  the  solstice  until  the 
22d  of  June,  that  is,  it  would  have  been  behind  its  time.     At  the 
end  of  the  next  four  years  the  solstice  would  fall  on  the  23d ; 
and  in  process  of  time  it  would  fall  successively  on  every  day  of 
the   year.     The  same  would  be  true  of  any  other  fixed   date. 
Julius  Caesar  made  the  first  correction  of  the  calendar,  by  intro- 
ducing an  intercalary  day  every  fourth  year,  making  February 
to  consist  of  29  instead  of  28  days,  and  of  course  the  whole  year 
to  consist  of  366  days.     This  fourth  year  was  denominated  Bis- 
sextile.*,   It  is  also  called  Leap  Year. 

112.  But  the  true  correction  was  not  6  hours,  but  5h.  49m.; 
hence  the  intercalation  was  too  great  by  1 1  minutes.     This  small 
fraction  would  amount  in  100  years  to  £  of  a  day,  and  in  1000 

*  The  sextus  dies  ante  Kalendas  being  reckoned  twice,  (Bis). 


46  THE    EARTH. 

years  to  more  than  7  days.  From  the  year  325  to  1582,  it  had 
in  fact  amounted  to  about  10  days;  for  it  was  known  that  in  325, 
the  vernal  equinox  fell  on  the  21st  of  March,  whereas,  in  1582  it 
fell  on  the  llth.  In  order  to  restore  the  equinox  to  the  same  date, 
Pope  Gregory  XIII  decreed,  that  the  year  should  be  brought  for- 
ward ten  days,  by  reckoning  the  5th  of  October  the  15th.  In  or- 
der to  prevent  the  calendar  from  falling  into  confusion  afterwards, 
the  following  rule  was  adopted : 

Every  year  whose  number  is  not  divisible  by  4  without  a  re- 
mainder, consists  of  365  days  ;  every  year  which  is  so  divisible,  but 
is  not  divisible  by  100,  of  366 ;  every  year  divisible  by  100  but  not 
by  400,  again  of  365  ;  and  every  year  divisible  by  400,  of  366. 

Thus  the  year  1838,  not  being  divisible  by  four,  contains  365  days, 
while  1836  and  1840  are  leap  years.  Yet  to  make  every  fourth 
year  consist  of  366  days  would  increase  it  too  much  by  about  £ 
of  a  day  in  100  years ;  therefore  every  hundredth  year  has  only 
365  days.  Thus  1800,  although  divisible  by  4,  was  not  a  leap 
year,  but  a  common  year.  But  we  have  allowed  a  whole  day 
in  a  hundred  Jrears,  whereas  we  ought  to  have  allowed  only  three 
fourths  of  a  day.  Hence,  in  400  years  we  should  allow  a  day  too 
much,  and  therefore  we  let  the  400th  year  remain  a  leap  year. 
This  rule  involves  an  error  of  less  than  a  day  in  4237  years.*  If 
the  rule  were  extended  by  making  every  year  divisible  by  4,000 
(which  would  now  consist  of  366  days)  to  consist  of  365  days,  the 
error  would  not  be  more  than  one  day  in  100,000  years. f 

113.  This  reformation  of  the  calendar  was  not  adopted  in  Eng- 
land until  1752,  by  which  time  the  error  in  the  Julian  calendar 
amounted  to  about  11  days.  The  year  was  brought  forward,  by 
reckoning  the  3d  of  September  the  14th.  Previous  to  that  time 
the  year  began  the  25th  of  March ;  but  it  was  now  made  to  be- 
gin on  the  1st  of  January,  thus  shortening  the  preceding  year, 
1751,  one  quarter.  J 


*  Woodhouse,  p.  874.  t  Herscliel's  Ast.  p.  384. 

\  Russia,  and  the  Greek  Church  generally,  adhere  to  the  old  style.  In  order  to  make 
the  Russian  dates  correspond  to  ours,  we  must  add  to  them  12  days.  France  and  other 
Catholic  countries,  adopted  the  Gregorian  calendar  soon  after  it  was  promulgated. 


THE    CALENDAR.  47 

114.  As  in  the  year  1582,  the  error  in  the   Julian  calendar 
amounted  to  10  days,  and  increased  by  £  of  a  day  in  a  century, 
at  present  the  correction  is  12  days ;  and  the  number  of  the  year 
will  differ  by  one  with  respect  to  dates  between  the  1st  of  Janu- 
ary and  the  25th  of  March. 

Examples.  General  Washington  was  born  Feb.  11,  1731,  old 
style  ;  to  what  date  does  this  correspond  in  new  style  ? 

As  the  date  is  the  earlier  part  of  the  18th  century,  the  correc- 
tion is  1 1  days,  which  makes  the  birth  day  fall  on  the  22d  of 
February ;  and  since  the  year  1731  closed  the  25th  of  March, 
while  according  to  new  style  1732  would  have  commenced  on 
the  preceding  1st  of  January ;  therefore,  the  time  required  is  Feb. 
22,  1732.  It  is  usual,  in  such  cases,  to  write  both  years,  thus- 
Feb.  11,1731-2,0.8. 

2.  A  great  eclipse  of  the  sun  happened  May  15th,  1836 ;  to 
what  date  would  this  time  correspond  in  old  style  ? 

Ans.   May.  3d, 

115.  The  common  year  begins  and  ends  on  the  same  day  of 
the  week ;  but  leap  year  ends  one  day  later  in  the  week  than  it  began. 

For  52x7=364  days;  if  therefore  the  year  begins  on  Tues- 
day, for  example,  364  days  would  complete  52  weeks,  and  one 
day  would  be  left  to  begin  another  week,  and  the  following  year 
would  begin  on  Wednesday.  Hence,  any  day  of  the  month  is  one 
day  later  in  the  week  than  the  corresponding  day  of  the  preceding 
year.  Thus,  if  the  16th  of  November,  1838,  falls  on  Friday, 
the  16th  of  November,  1837,  fell  on  Thursday,  and  will  fall  in 
1839  on  Saturday.  But  if  leap  year  begins  on  Sunday,  it  ends 
on  Monday,  and  the  following  year  begins  on  Tuesday ;  while 
any  given  day  of  the  month  is  two  days  later  in  the  week  than 
the  corresponding  date  of  the  preceding  year. 

116.  Fortunately  for  astronomy,  the  confusion  of  dates  involved 
in  different  calendars  affects  recorded  observations  but  little.     Re- 
markable eclipses,  for  example,  can  be  calculated  back  for  several 
thousand  years,  without  any  danger  of  mistaking  the  day  of  their 
occurrence  ;  and  whenever  any  such  eclipse  is  so  interwoven  with 
the  account  given  by  an  ancient  author  of  s6me  historical  event, 


48  THE    EARTH. 

as  to  indicate  precisely  the  interval  of  time  between  the  eclipse 
and  the  event,  and  at  the  same  time  completely  to  identify  the 
eclipse,  that  date  is  recovered  and  fixed  forever.* 


CHAPTER  V. 

OF  ASTRONOMICAL   INSTRUMENTS  AND    PROBLEMS FIGURE    AND  DEN 

SITY    OF    THE    EARTH. 

117.  THE  most  ancient  astronomers  employed  no  instruments 
for  measuring  angles,  but  acquired  their  knowledge  of  the  heav- 
enly bodies  by  long  continued  and  most  attentive  inspection  with 
the  naked  eye.  In  the  Alexandrian  school,  about  300  years  before 
the  Christian  era,  instruments  began  to  be  freely  used,  and  thence- 
forward trigonometry  lent  a  powerful  aid  to  the  science  of  astron- 
omy. Tycho  Brahe,  in  the  16th  century,  formed  a  new  era  in 
poetical  astronomy,  and  carried  the  measurement  of  angles  to 
10 ', — a  degree  of  accuracy  truly  wonderful,  considering  that  he 
had  not  the  advantage  of  the  telescope.  By  the  application  of 
the  telescope  to  astronomical  instruments,  a  far  better  defined  view 
of  objects  was  acquired,  and  a  far  greater  degree  of  refinement 
was  attainable.  The  astronomers  royal  of  Great  Britain  perfected 
the  art  of  observation,  bringing  the  measurement  of  angles  to  1", 
and  the  estimation  of  differences  of  time  to  TV  of  a  second.  Be- 
yond this  degree  of  refinement  it  is  supposed  that  we  cannot 
advance,  since  unavoidable  errors  arising  from  the  uncertainties 
of  refraction,  and  the  necessary  imperfection  of  instruments,  for- 
bid us  to  hope  for  a  more  accurate  determination  than  this.  But 
a  little  reflection  will  show  us,  that  I"  on  the  limb  of  an  astro- 
nomical instrument,  must  be  a  space  exceedingly  small.  Suppose 
the  circle,  on  which  the  angle  is  measured,  be  one  foot  in  diameter. 


*  An  elaborate  view  of  the  Calendar  may  be  found  in  Delambre's  Astronomy,  t.  III. 
A  useful  table  for  finding  the  day  of  the  week  of  any  given  date,  is  inserted  in  the 
American  Almanac  for  1832,  p.  72. 


ASTRONOMICAL    INSTRUMENTS.  49 

Then  -12X3'14159=Ty  inch  =  space   occupied  by   1°.     Hence 
360 

= =space  of  T,  and  ^-,-7;77=sPace  of  1".    Such  minute 


10x60     600  36000 

angles  can  be  measured  only  by  large  circles.  If,  for  example, 
a  circle  is  20  feet  in  diameter,  a  degree  on  its  periphery  would 
occupy  a  space  20  times  as  large  as  a  degree  on  a  circle  of  1  foot. 
A  degree  therefore  of  the  limb  of  such  an  instrument  would 
occupy  a  space  of  2  inches :  one  minute,  gV  of  an  inch  ;  and  one 
second,  TJVo-  of  an  inch. 

118.  But  the  actual  divisions  on  the  limb  of  an  astronomical 
instrument  never  extend  to  seconds :  in  the  smaller  instruments 
they  reach  only  to  10',  and  on  the  largest  rarely  lower  than  1'. 
The  subdivision  of  these  spaces  is  carried  on  by  means  of  the 
Vernier,  which  may  be  thus  defined : 

A  VERNIER  is  a  contrivancce  attached  to  the  graduated  limb  of 
an  instrument,  for  the  purpose  of  measuring  aliquot  parts  of  the 
smallest  spaces,  into  which  the  instrument  is  divided. 

The  vernier  is  usually  a  narrow  zone  of  metal,  which  is  made 
to  slide  on  the  graduated  limb.  Its  divisions  correspond  to  those 
on  the  limb,  except  that  they  are  a  little  larger,*  one  tenth,  for 
example,  so  that  ten  divisions  on  the  vernier  would  equal  eleven 
on  the  limb.  Suppose  now  that  our  instrument  is  graduated  to 
degrees  only,  but  the  altitude  of  a  certain  star  is  found  to  be  40° 
and  a  fraction,  or  40°  +x.  In  order  to  estimate  the  amount  of  this 
fraction,  we  bring  the  zero  point  of  the  vernier  to  coincide  with 
the  point  which  indicates  the  exact  altitude,  or  40°  +x.  We  then 
look  along  the  vernier  until  we  find  where  one  of  its  divisions 
coincides  with  one  of  the  divisions  of  the  limb.  Let  this  be  at  the 
fourth  division  of  the  vernier.  In  four  divisions,  therefore,  the  ver- 
nier has  gained  upon  the  divisions  of  the  limb,  a  space  equal  to  x ; 
and  since,  in  the  case  supposed,  it  gains  Ty  of  a  degree,  or  6'  at  each 
division,  the  entire  gain  is  24',  and  the  arc  in  question  is  40°  24'. 

119.  As  the  vernier  is  used  in  the  barometer,  where  its  applica- 

*  In  the  more  modern  instruments  the  divisions  of  the  vernier  are  smaller  than  those 
of  the  limb 

1 


50 


THE    EARTH. 


Fig.  13. 


-1—31 


-30 


-29 


tion  is  more  easily  seen  than  in  astronomical  instruments,  while  the 
principle  is  the  same  in  both  cases,  let  us 
see  how  it  is  applied  to  measure  the  ex- 
act height  of  a  column  of  mercury.  Let 
AB  (Fig.  13,)  represent  the  upper  part 
of  a  barometer,  the  level  of  the  mercury 
being  at  C,  namely,  at  30.3  inches,  and 
nearly  another  tenth.  The  vernier  being 
brought  (by  a  screw  which  is  usually  at- 
tached to  it)  to  coincide  with  the  surface 
of  the  mercury,  we  look  along  down  the 
scale,  until  we  find  that  the  coincidence 
is  at  the  8th  division  of  the  vernier. 
Now  as  the  vernier  gains  TV  of  TV=T^o- 
of  an  inch  at  each  division  upward,  it  of 
course  gains  TJT  in  eight  divisions.  The  fractional  quantity,  there- 
fore, is  .08  of  an  inch,  and  the  height  of  the  mercury  is  30.38.  If 
the  divisions  of  the  vernier  were  such,  that  each  gained  g-V  (when 
60  on  the  vernier  would  equal  61  on  the  limb)  on  a  limb  divided 
into  degrees,  we  could  at  once  take  off  minutes  ;  and  were  the  limb 
graduated  to  minutes,  we  could  in  a  similar  way  read  off  seconds. 

120.  The  instruments  most  used  for  astronomical  observations, 
are  the  Transit  Instrument,  the  Astronomical  Clock,  the  Mural 
Circle,  and  the  Sextant.  A  large  portion  of  all  the  observations, 
made  in  an  astronomical  observatory,  are  taken  on  the  meridian. 
When  a  heavenly  body  is  on  the  meridian,  being  at  its  highest 
point  above  the  horizon,  it  is  then  least  affected  by  refraction  and 
parallax  ;  its  zenith  distance  (from  which  its  altitude  and  decli- 
nation are  easily  derived)  is  readily  estimated ;  and  its  right  as- 
cension may  be  very  conveniently  and  accurately  determined  by 
means  of  the  astronomical  clock.  Having  the  right  ascension 
and  declination  of  a  heavenly  body,  various  other  particulars  re- 
specting its  position  may  be  found,  as  we  shall  see  hereafter,  by 
the  aid  of  spherical  trigonometry.  Let  us  then  first  turn  our  at- 
tention to  the  instruments  employed  for  determining  the  right 
ascension  and  declination.  They  are  the  Transit  Instrument,  the 
Astronomical  Clock,  and  the  Mural  Circle. 


ASTRONOMICAL    INSTRUMENTS. 


51 


121.  The  Transit  Instrument  is  a  telescope,  which  is  fixed 
permanently  in  the  meridian,  and  moves  only  in  that  plane.  It 
rests  on  a  horizontal  axis,  which  consists  of  two  hollow  cones 
applied  base  to  base,  a  form  uniting  lightness  and  strength.  The 
two  ends  of  the  axis  rest  on  two  firm  supports,  as  pillars  of  stone, 
for  example,  usually  built  up  from  the  ground,  and  so  related  to 
the  building  as  to  be  as  free  as  possible  from  all  agitation.  In 
figure  14,  AD  represents  the  telescope,  E,  W,  massive  stone  pillars 
supporting  the  horizontal  axis,  beneath  which  is  seen  a  spirit  level, 
(which  is  used  to  bring  the  axis  to  a  horizontal  position,)  and  n  a 
vertical  circle  graduated  into  degrees  and  minutes.  This  circle 
serves  the  purpose  of  placing  the  instrument  at  any  required  alti- 
tude or  distance  from  the  zenith,  and  of  course  for  determining 
altitudes  and  zenith  distances. 

Fig.  14. 


122.  Various  methods  are  described  in  works  on  practical  as- 
tronomy, for  placing  the  Transit  Instrument  accurately  in  the 
meridian.  The  following  method  by  observations  on  the  pole 
star,  may  serve  as  an  example  If  the  instrument  be  directed 


52 


THE   EARTH. 


towards  the  north  star,  and  so  adjusted  that  the  star  Alioth  (the 
first  in  the  tail  of  the  Great  Bear)  and  the  pole  star  are  both  in 
the  same  vertical  circle,  the  former  below  the  pole  and  the  latter 
above  it,  the  instrument  will  be  nearly  in  the  plane  of  the  meridian. 
To  adjust  it  more  exactly,  compare  the  time  occupied  by  the  pole 
star  in  passing  from  its  upper  to  its  lower  culmination,  with  the 
time  of  passing  from  its  lower  to  its  upper  culmination.  These 
two  intervals  ought  to  be  precisely  equal ;  and  if  they  are  so,  the 
iustrument  is  truly  placed  in  the  meridian  ;  but  if  they  are  not 
equal,  the  position  of  the  instrument  must  be  shifted  until  they 
become  exactly  equal. 

123.  The  line  of  collimation  of  a  telescope,  is  a  line  joining  the 
center  of  the  object  glass  with  the  center  of  the  eye  glass.  When 
the  transit  instrument  is  properly  adjusted,  this  line,  as  the  instru- 
ment is  turned  on  its  axis,  moves  in  the  plane  of  the  meridian. 
Having,  by  means  of  the  vertical  circle  n,  set  the  instrument  at 
the  known  altitude  or  zenith  distance  of  any  star,  upon  which  we 
wish  to  make  observations,  we  wait  until  the  star  enters  the  field 
of  the  telescope,  and  note  the  exact  instant  when  it  crosses  the 
vertical  wire  in  the  center  of  the  field,  which  wire  marks  the  true 
plane  of  the  meridian.  Usually,  however,  there  are  placed  in  the 
focus  of  the  eye  glass  five  parallel  wires  or  threads,  two  on  each 
side  of  the  central  wire,  and  all 
at  equal  distances  from  each 
other,  as  is  represented  in  the 
following  diagram.  The  time 
of  arriving  at  each  of  the  wires 
being  noted,  and  all  the  times 
added  together  and  divided  by 
the  number  of  observations,  the 
result  gives  the  instant  of  cross- 
ing the  central  wire. 


124.  The  Astronomical  Clock 
is  the  constant  companion  of  the 
Transit  Instrument.     This  clock  is  so  regulated  as  to  keep  exact 
pace  with  the  stars,  and  of  course  with  the  revolution  of  the  earth 


ASTRONOMICAL    INSTRUMENTS.  53 

on  its  axis  ;  that  is,  it  is  regulated  to  sidereal  time.  It  measures 
the  progress  of  a  star,  indicating  an  hour  for  every  15°,  and  24 
hours  for  the  whole  period  of  the  revolution  of  the  star.  Sidereal 
time,  it  will  be  recollected,  commences  when  the  vernal  equinox 
is  on  the  meridian,  just  as  solar  time  commences  when  the  sun  is 
on  the  meridian.  Hence,  the  hour  by  the  sidereal  clock  has  no 
correspondence  with  the  hour  of  the  day,  but  simply  indicates 
how  long  it  is  since  the  equinoctial  point  crossed  the  meridian. 
For  example,  the  clock  of  an  observatory  points  to  3h.  20m. ;  this 
may  be  in  the  morning,  at  noon,  or  any  other  time  of  the  day,  since 
it  merely  shows  that  it  is  3h.  20m.  since  the  equinox  was  on  the 
meridian.  Hence,  when  a  star  is  on  the  meridian,  the  clock 
itself  shows  its  right  ascension  ;  and  the  interval  of  time  between 
the  arrival  of  any  two  stars  upon  the  meridian,  is  the  measure  of 
their  difference  of  right  ascension. 

125.  Astronomical  clocks  are  made  of  the  best  workmanship, 
with  a  compensation  pendulum,  and  every  other  advantage  which 
can  promote  their  regularity.  The  Transit  Instrument  itself, 
when  once  accurately  placed  in  the  meridian,  affords  the  means 
of  testing  the  correctness  of  the  clock,  since  one  revolution  of  a 
star  from  the  meridian  to  the  meridian  again,  ought  to  correspond 
to  exactly  24  hours  by  the  clock,  and  to  continue  the  same  from 
day  to  day  ;  and  the  right  ascension  of  various  stars,  as  they  cross 
the  meridian,  ought  to  be  such  by  the  clock  as  they  are  given  in 
the  tables,  where  they  are  stated  according  to  the  most  accurate 
determinations  of  astronomers.  Or  by  taking  the  difference  of 
right  ascension  of  any  two  stars  on  successive  days,  it  will  be  seen 
whether  the  going  of  the  clock  is  uniform  for  that  part  of  the 
day ;  and  by  taking  the  right  ascension  of  different  pairs  of  stars, 
we  may  learn  the  rate  of  the  clock  at  various  parts  of  the  day. 
We  thus  learn,  not  only  whether  the  clock  accurately  measures 
the  length  of  the  sidereal  day,  but  also  whether  it  goes  uniformly 
from  hour  to  hour. 

Although  astronomical  clocks  have  been  brought  to  a  great  de- 
gree of  perfection,  so  as  to  vary  hardly  a  second  for  many  months, 
yet  none  are  absolutely  perfect,  and  most  are  so  far  from  it  as  to 
require  to  be  corrected  by  means  of  the  Transit  Instrument  every 


54  THE    EARTH. 

few  days.  Indeed,  for  the  nicest  observations,  it  is  usual  not  t« 
attempt  to  bring  the  clock  to  an  absolute  state  of  correctness,  but 
after  bringing  it  as  near  to  such  a  state  as  can  conveniently  be 
done,  to  ascertain  how  much  it  gains  or  loses  in  a  day ;  that  is,  to 
ascertain  its  rate  of  going,  and  to  make  allowance  accordingly. 

126.  The  vertical  circle  (n,  Fig.  14,)  usually  connected  with 
the  Transit  Instrument,  affords  the  means  of  measuring  arcs  on 
the  meridian,  as  meridian  altitudes,  zenith  distances,  and  decli- 
nations ;  but  as  the  circle  must  necessarily  be  small,  and  there- 
fore incapable  of  measuring  very  minute  angles,  the  Mural  Cir- 
cle is  usually  employed  for  measuring  arcs  of  the  meridian.  The 
Mural  Circle  is  a  graduated  circle,  usually  of  very  large  size,  fixed 
permanently  in  the  plane  of  the  meridian,  and  attached  firmly  to 
a  perpendicular  wall.  It  is  made  of  large  size,  sometimes  1 1  feet 
in  diameter,  in  order  that  very  small  angles  may  be  measured  on 
its  limb ;  and  it  is  attached  to  a  massive  wall  of  solid  masonry  in 
order  to  insure  perfect  steadiness,  a  point  the  more  difficult  to 
attain  in  proportion  as  the  instrument  is  heavier.  The  annexed 
diagram  represents  a  Mural  Circle  fixed  to  its  wall  and  ready  for 
observations.  It  will  be  seen  that  every  expedient  is  employed 
to  give  the  instrument  firmness  of  parts  and  steadiness  of  position. 
Its  radii  are  composed  of  hollow  cones,  uniting  lightness  and 
strength,  and  its  telescope  revolves  on  a  large  horizontal  axis, 
fixed  as  firmly  as  possible  in  a  solid  wall.  The  graduations  are 
made  on  the  outer  rim  of  the  instrument,  and  are  read  off  by  six 
microscopes  (called  reading  microscopes)  attached  to  the  wall,  one 
of  which  is  represented  at  A,  and  the  places  of  the  five  others 
are  marked  by  the  letters  B,  C,  D,  E,  F.  Six  are  used,  in  order 
that  by  taking  the  mean  of  such  a  number  of  readings,  a  higher 
degree  of  accuracy  may  be  insured,  than  could  be  obtained  by  a 
single  reading.  In  a  circle  of  six  feet  diameter,  like  that  repre- 
sented in  the  figure,  the  divisions  may  be  easily  carried  to  five 
minutes  each.  The  microscope  (which  is  of  the  variety  called 
compound  microscope)  forms  an  enlarged  image  of  each  of  these 
divisions  in  the  focus  of  the  eye  glass.  With  it  is  combined  the 
principle  of  the  micrometer.  This  is  effected  by  placing  in  the 
focus  a  delicate  wire,  which  may  be  moved  by  means  of  a  screw 


ASTRONOMICAL    INSTRUMENTS. 

Fig.  16. 


55 


m  a  direction  parallel  to  the  divisions  of  the  limb,  and  which  is  so 
adjusted  to  the  screw  as  to  move  over  the  whole  magnified  space 
of  five  minutes  by  five  revolutions  of  the  screw.  Of  course  one 
revolution  of  the  screw  measures  one  minute.  Moreover,  if  the 
screw  itself  is  made  to  carry  an  index  attached  to  its  axis  and  re- 
volving with  it  over  a  disk  graduated  into  sixty  equal  parts,  then 
the  space  measured  by  moving  the  index  over  one  of  these  parts, 
will  be  one  second. 

We  have  been  thus  minute  in  the  description  of  this  instrument, 
in  order  to  give  the  learner  some  idea  of  the  vast  labor  and  great 
patience  demanded  of  practical  astronomers,  in  order  to  obtain 
measurements  of  such  extreme  accuracy  as  those  to  which  they 
aspire. 

On  account  of  the  great  dimensions  of  this  circle,  and  the  ex- 
pense attending  it,  as  well  as  the  difficulty  of  supporting  it  firmly, 
sometimes  only  one  fourth  of  it  is  employed,  constituting  the  Mu- 
ral Quadrant.  This  instrument  has  the  disadvantage,  however, 


56 


THE    EARTH. 


of  being  applicable  to  only  one  hemisphere  at  a  time,  either  the 
northern  or  the  southern,  according  as  it  is  fixed  to  the  eastern 
or  the  western  side  of  the  wall. 


127.  We  have  before  shown  (Art.  124,)  the  method  of  finding 
the  right  ascension  of  a  star  by  means  of  the  Transit  Instrument 
and  the  clock.     The  declination  may  be  obtained  by  means  of  the 
mural  circle  in  several  different  ways,  our  object  being  always  to 
find  the  distance  of  the  star,  when  on  the  meridian,  from  the  equa- 
tor (Art.  37.)     First,  the  declination  may  be  found  from  the  me- 
ridian altitude.     Let  S  (Fig.  17,)  be  the  place  of  a  star  when 
on  the  meridian.     Then  its  meridian  altitude  will  be  SH,  which 
will  best  be  found  by  taking  its  ze- 
nith distance  ZS,  of  which  it  is  the 

complement.  From  SH,  subtract  EH, 
the  elevation  of  the  equator,  which 
equals  the  co-latitude  of  the  place  of 
observation,  (Art.  44,)  and  the  remain- 
der SE  is  the  declination.  Or  if  the 
star  is  nearer  the  horizon  than  the 
equator  is,  as  at  S',  subtract  its  me- 
ridian altitude  from  the  co-latitude,  for 
the  declination.  Secondly,  the  declination  may  be  found  from 
the  north  polar  distance,  of  which  it  is  the  complement.  Thus 
from  P  to  E  is  90°.  Therefore,  PE-PS=90°-PS=SE=the 
decimation.  The  height  of  the  pole  P  is  always  known  when  the 
latitude  of  the  place  is  known,  being  equal  to  the  latitude. 

128.  The  astronomical  instruments  already  described  are  adapt- 
ed to  taking  observations  on  the  meridian  only ;  but  we  some- 
times require  to  know  the  altitude  of  a  celestial  body  when  it  is 
not  on  the  meridian,  and  its  azimuth,  or  distance  from  the  meridian 
measured  on  the  horizon ;  and  also  the  angular  distance  between 
two  points  on  any  part  of  the  celestial  sphere.     An  instrument 
especially  designed  to  measure  altitudes  and  azimuths,  is  called  an 
Altitude  and  Azimuth  Instrument,  whatever  may  be  its  particular 
form.     When  a  point  is  on  the  horizon  its  distance  from  the  me- 
ridian, or  its  azimuth,  may  be  taken  by  the  common  surveyor's 


ASTRONOMICAL   INSTRUMENTS.  57 

compass,  the  direction  of  the  meridian  being  determined  by  the 
needle ;  but  when  the  object,  as  a  star,  is  not  on  the  horizon,  its 
azimuth,  it  must  be  remembered,  is  the  arc  of  the  horizon  froir 
the  meridian  to  a  vertical  circle  passing  through  the  star  (Art.  27)  ; 
at  whatever  different  altitudes,  therefore,  two  stars  may  be,  and 
however  the  plane  which  passes  through  them  may  be  inclined  to 
the  horizon,  still  it  is  their  angular  distance  measured  on  the  hori- 
zon which  determines  their  difference  of  azimuth.  Figure  18  rep- 
resents an  Altitude  and  Azimuth  Instrument,  several  of  the  usual 
appendages  and  subordinate  contrivances  being  omitted  for  the 
sake  of  distinctness  and  simplicity.  Here  abc  is  the  vertical  or 
altitude  circle,  and  EFG  the  horizontal  or  azimuth  circle  ;  AB  is  a 

Fig.  18. 


telescope  mounted  on  a  horizontal  axis  and  capable  of  two  mo- 
tions, one  in  altitude  parallel  to  the  circle  abc,  and  the  other  in 
azimuth  parallel  to  EFG.  Hence  it  can  be  easily  brought  to  bear 
upon  any  object.  At  m,  under  the  eye  glass  of  the  telescope,  is  a 
small  mirror  placed  at  an  angle  of  45°  with  the  axis  of  the  tele- 
scope, by  means  of  which  the  image  of  the  object  is  reflected  up- 
wards, so  as  to  be  conveniently  presented  to  the  eye  of  the  ob- 

8 


58  THE     EARTH. 

server.  At  d  is  represented  a  tangent  screw,  by  which  a  slow 
motion  is  given  to  the  telescope  at  c.  At  h  and  g  are  seen  two 
spirit  levels  at  right  angles  to  each  other,  which  show  when  the 
azimuth  circle  is  truly  horizontal.  The  instrument  is  supported 
on  a  tripod,  for  the  sake  of  greater  steadiness,  each  foot  being 
furnished  with  a  screw  for  levelling. 

129.  The  sextant  is  one  of  the  most  useful  instruments,  both 
to  the  astronomer  and  the  navigator,  and  will  therefore  merit 
particular  attention.  In  figure  19, 1  and  H  are  two  small  mirrors, 
and  T  a  small  telescope.  I  D  represents  a  movable  arm,  or 
radius,  which  carries  an  index  at  D.  The  radius  turns  on  a  pivot 
at  I,  and  the  index  moves  on  a  graduated  arc  EF.  I  is  called 

Fig.  19. 


the  Index  Glass  and  H  the  Horizon  Glass.  The  under  part  only 
of  the  horizon  glass  is  coated  with  quicksilver,  the  upper  part 
being  left  transparent ;  so  that  while  one  object  is  seen  through 
the  upper  part  by  direct  vision,  another  may  be  seen  through 
the  lower  part  by  reflexion  from  the  two  mirrors.  The  instru- 
ment is  so  contrived,  that  when  the  index  is  moved  up  to  F, 
where  the  zero  point  is  placed,  or  the  graduation  begins,  the  two 


ASTRONOMICAL    INSTRUMENTS.  59 

reflectors  I  and  H  are  exactly  parallel  to  each  other.  If  w« 
now  look  through  the  telescope,  T,  so  pointed  as  to  see  the  star 
S  through  the  transparent  part  of  the  horizon  glass,  we  shall 
see  the  same  star,  in  the  same  place,  reflected  from  the  silvered 
part ;  for  the  star  (or  any  similar  object)  is  at  such  a  distance 
that  the  rays  of  light  which  strike  upon  the  index  glass  I,  are 
parallel  to  those  which  enter  the  eye  directly,  and  will  exhibit 
the  object  at  the  same  place.  Now,  suppose  we  wish  to  meas- 
ure the  angular  distance  between  two  bodies,  as  the  moon  and  a 
star,  and  let  the  star  be  at  S  and  the  moon  at  M.  The  telescope 
being  still  directed  to  S,  turn  the  index  arm  I D  from  F  towards 
E  until  the  image  of  the  moon  is  brought  down  to  S,  its  lower 
limb  just  touching  S.  By  a  principle  in  optics,  the  angular  dis- 
tance which  the  image  of  the  moon  passes  over,  is  twice  that  of 
the  mirror  I.  But  the  mirror  has  passed  over  the  graduated  arc 
FD  ;  therefore  double  that  arc  is  the  angular  distance  between 
the  star  and  the  moon's  lower  limb.  If  we  then  bring  the  upper 
limb  into  contact  with  the  star,  the  sum  of  both  observations, 
divided  by  2,  will  give  the  angular  distance  between  the  star 
and  the  moon's  center.  As  each  degree  on  the  limb  EF  meas- 
ures two  degrees  of  angular  distance,  hence  the  divisions  for  sin- 
gle degrees  are  in  fact  only  half  a  degree  asunder ;  and  a  sextant, 
or  the  sixth  part  of  the  circle,  measures  an  angular  distance  of 
120°.  The  upper  and  lower  points  in  the  disk  of  the  sun  or  of 
the  moon,  may  be  considered  as  two  separate  objects,  whose 
distance  from  each  other  may  be  taken  in  a  similar  manner, 
and  thus  their  apparent  diameters  at  any  time  be  determined. 
We  may  select  our  points  of  observation  either  in  a  vertical,  or 
in  a  horizontal  direction. 

130.  If  we  make  a  star,  or  the  limb  of  the  sun  or  moon,  one  of 
the  objects,  and  the  point  in  the  horizon  directly  beneath,  the  oth- 
er, we  thus  obtain  the  altitude  of  the  object.  In  this  observation, 
the  horizon  is  viewed  through  the  transparent  part  of  the  hori- 
zon glass.  At  sea,  where  the  horizon  is  usually  well  defined,  the 
horizon  itself  may  be  used  for  taking  altitudes  ;  but  on  land,  in- 
equalities of  the  earth's  surface,  oblige  us  to  have  recourse  to  an 
artificial  horizon.  This,  in  its  simple  state,  is  a  basin  of  either 


60  THE    EARTH. 

water  or  quicksilver.  By  this  means  we  see  the  image  of  the 
sun  (or  other  body)  just  as  far  below  the  horizon  as  it  is  in  reality 
above  it.  Hence,  if  we  turn  the  index  glass  until  the  limb  of  the 
sun,  as  reflected  from  it,  is  brought  into  contact  with  the  image 
seen  in  the  artificial  horizon,  we  obtain  double  the  altitude.* 

The  sextant  must  be  held  in  such  a  manner,  that  its  plane  shall 
pass  through  the  plane  of  the  two  objects.  It  must  be  held 
therefore  in  a  vertical  plane  in  taking  altitudes,  and  in  a  horizontal 
plane  in  taking  the  horizontal  diameters  of  the  sun  and  moon. 
Holding  the  instrument  in  the  true  plane  of  the  two  bodies,  whose 
angular  distance  is  measured,  is  indeed  the  most  difficult  part  of 
the  operation. 

The  peculiar  value  of  the  sextant  consists  in  this,  that  the  ob- 
servations taken  with  it  are  not  affected  by  any  motion  in  the 
observer ;  hence  it  is  the  chief  instrument  used  for  angular  meas- 
urements at  sea. 

131.  Examples  illustrating  the  use  of  the  Sextant. 
Ex.  1.  Alt.  0's  lower  limb,          .         .        49°  10'  00" 
©'s  semi-diameter,    .     .    ,          0    15  51 

49°  25'  51" 
Subtract  Refraction,       .        .        00    00  49 


49°  25'  02" 
Add  Parallax,         ...        00    00  06 


True  altitude  0's  center,          .        49°  25'  08" 

Ex.  2.  With  the  Artificial  Horizon. 

Altitude  of  0's  upper  limb  above  the  image  in  the  artificial  ho- 
rizon, 100°  2'  47". 

True  altitude, 50°  01' 23."5 

©'s  semi-diameter,         .        .        .  00    15  50. 

49°  45'  33."5 
Refraction, 00    00  48. 

49°  44'  45."5 
Parallax, 00   00  05. 

True  altitude  of  ©'s  center,  .        .        .        49°  44' 5Q."5 

*  Woodhouse's  Ast.  p.  774. 


ASTRONOMICAL  PROBLEMS.  61 

ASTRONOMICAL  PROBLEMS.* 

132.  Given  the  sun's  Right  Ascension  and  Declination,  to  find 
his  Longitude  and  the  Obliquity  of  the  Ecliptic. 

Let  PCP'  (Fig.  20,)  represent  the  celestial  meridian  that  passes 
through  the  first  of  Cancer  and  Capricorn,  (the  solstitial  colure,) 
PP'  the  axis  of  the  sphere,  EQ  the  equator,  E'C  the  ecliptic,  and 
PSP'  the  declination  circle  (Art. 
37,)  passing  through  the  sun  S ; 
then  ARS  is  a  right  angle,  and  in 
the  right  angled  spherical  triangle 
ARS,  are  given  the  right  ascension 
AR  (Art.  37,)  and  the  declination 
RS,  to  find  the  longitude  AS  and 
the  obliquity  SAR. 

As  longitude  and  right  ascension 
are  measured  from  A,  the  first  point 

of  Aries,  in  the  direction  AS  of  the  signs,  quite  round  the  globe, 
when,  of  the  four  quantities  mentioned  in  the  problem,  the  obliquity 
and  the  declination  are  given  to  find  the  others,  we  must  know 
whether  the  sun  is  north,  or  whether  it  is  south  of  the  equator,  the 
longitude  being  in  the  one  case  AS,  and  in  the  other,  instead  of 
AS',  it  is  360 —AS',  that  is,  the  supplement  of  AS'.  We  must 
also  know  on  which  side  of  the  tropic  the  sun  is,  for  the  sun  in 
passing  from  one  of  the  tropics  to  the  equinox,  passes  through  the 
same  degrees  of  declination  as  it  had  gone  through  in  ascending 
from  the  other  equinox  to  the  tropic,  although  its  longitude  and 
right  ascension  go  on  continually  increasing.  From  the  21st  of 
March  to  the  21st  of  June,  while  describing  the  first  quadrant 
from  the  vernal  equinox,  the  declination  is  north  and  increasing ; 
north  but  decreasing,  in  the  second  quadrant,  until  the  23d  of 
September ;  south  and  increasing  in  the  third  quadrant,  until  the 
21st  of  December ;  and  finally,  in  the  fourth  quadrant,  south  but 
decreasing  until  the  21st  of  March. 

Ex.  1.  On  the  17th  of  May,  the  sun's  Right  Ascension  was 
53°  38',  and  his  Declination  19°  15'  57":  required  his  Longitude 
and  the  Obliquity  of  the  Ecliptic. 

»  Young's  Spherical  Trigonometry,  p.  136.    Vince's  Complete  System,  Vol.  I. 


62  THE  EARTH. 

Applying  Napier's  rule*  to  the  right  angled  triangle,  ARS,  we 
have 

1.  Rad.  cos.  AS=cos.  AR  cos.  RS. 

2.  Rad.  sin.  AR=tan.  RS  cot.  A. -.cot.  A=— — — 

tan.  RS 

Hence  the  computation  for  AS  and  A  is  as  follows : 

For  the  Longitude  AS.  For  the  Obliquity  A. 


cos.AR  53°  38'  00"     9.7730185 
cos.RS  19    15  57       9.9749710 


cos.AS  55    57  43       9.7479895 


sin  AR  9.9059247 

tan.  RS,  ar.  com.         0.4565209 


cot.  A  23°  27'  50£"    10.3624456 


Ex.  2.  On  the  31st  of  March,  1816,  the  sun's  Declination  was 
observed  at  Greenwich  to  be  4°  13'  3H":  required  his  Right 
Ascension,  the  obliquity  of  the  ecliptic  being  23°  27'  51". 

-  Ans.  9°  47'  59". 

Ex.  3.  What  was  the  sun's  Longitude  on  the  28th  of  Novem- 

*  The  student  *s  supposed  to  be  acquainted  with  Spherical  Trigonometry ;  but  to  re- 
fresh  his  memory,  we  may  insert  a  remark  or  two. 

It  will  be  recollected  that  in  Napier's  rule  for  the  solution  of  a  right  angled  spherical 
triangle,  by  means  of  the  Five  Circular  Parts,  we  proceed  as  follows.  £ 

In  a  right  angled  spherical  triangle  we  are  to  recognize  but  five  parts,  viz.  the  three 
sides  and  the  two  oblique  angles.  If  we  take  any  one  of  these  as  a  middle  part,  the 
two  which  lie  next  to  it  on  each  side  will  be  adjacent  parts.  Thus,  (in  Fig.  21,)  taking 
A  for  a  middle  part,  b  and  c  will  be  the  adjacent  parts ;  if  we  take  c  for  the  middle  part, 
A  and  B  will  be  the  adjacent  parts ;  if  we  fig.  21. 

take  B  for  the  middle  part,  c  and  a  will  be 
the  adjacent  parts ;  but  if  we  take  a  for 
the  middle  part,  then  as  the  angle  C  is 
not  considered  as  one  of  the  circular  parts, 
B  and  b  are  the  adjacent  parts ;  and,  last- 

ly,  if  b  is  the  middle  part,  then  the  adja-  ^ 

cent  parts  are  A  and  a.  The  two  parts  immediately  beyond  the  adjacent  parts  on  each 
side,  still  disregarding  the  right  angle,  are  called  the  opposite  parts ;  thus  if  A  is  the 
middle  part,  the  opposite  parts  are  a  and  B.  Napier's  rule  is  as  follows : 

Radius  into  the  sine  of  the  middle  part,  equals  the  product  of  the  tangents  of  the 
adjacent  extremes,  or  of  the  cosines  of  the  opposite  extremes. 

(The  corresponding  vowels  are  marked  to  aid  the  memory.)  This  rule  is  modified 
by  using  the  complements  of  the  two  angles  and  the  hypothenuse  instead  of  the  parts 
themselves.  Thus  instead  of  rad.Xsin.  A,  we  say  rad.Xcos.  A,  when  A  is  the  middle 
part ;  and  rad.Xcos.  AB,  when  the  hypothenuse  is  the  middle  part. 

Examples.  1.  In  the  right  angled  triangle  ABC,  are  given  the  two  perpendicular 
sides,  viz.  a=48°  24'  10",  6=59°  38'  27",  to  find  the  hypothenuse  c.  The  hypothenuse 
being  made  the  middle  part,  the  other  sides  become  the  opposite  parts,  being  separated 


ASTRONOMICAL  PROBLEMS.  63 

her,  1810,  when  his  Declination  was  21°  16'  4",  and  his  Right 
Ascension,  in  time,  16h.  14m.  58.4s.? 

Ans.  245°  39'  10". 

Ex.  4.  The  sun's  Longitude  being  8s.  7°  40'  56",  and  the  Ob- 
liquity 23°  2V  42i",  what  was  the  Right  Ascension  in  time? 

Ans.  16h.  23m.  34s. 

133.  Given  the  surfs  Declination  to  find  the  time  of  his  Rising 
and  Setting  at  any  place  whose  latitude  is  known. 

Let  PEP'  (Fig.  22,)  represent  the  meridian  of  the  place,  Z 
being  the  zenith,  and  HO  the  horizon ;  and  let  LL'  be  the  appa- 
rent path  of  the  sun  on  the  proposed 
day,  cutting  the  horizon  in  S.  Then 
the  arc  EZ  will  be  the  latitude  of  the 
place,  and  consequently  EH,  or  its 
equal  QO,  will  be  the  co-latitude,  and 
this  measures  the  angle  OAQ ;  also 
RS  will  be  the  sun's  declination,  and 
AR  expressed  in  time  will  be  the  time 
of  rising  before  6  o'clock.  For  it  is 
evident  that  it  will  be  sunrise  when 

the  sun  arrives  at  the  horizon  at  S  ;  but  PP'  being  an  hour  circle 
whose  plane  is  perpendicular  to  the  meridian,  (and  of  course  pro- 
jected into  a  straight  line  on  the  plane  of  projection,)  the  time  the 
sun  is  passing  from  S  to  S'  taken  from  the  time  of  describing  S'L, 
which  is  six  hours,  must  be  the  time  from  midnight  to  sunrise. 
But  the  time  of  describing  SS;  is  measured  on  the  corresponding 
arc  of  the  equinoctial  AR. 

In  the  right  angled  triangle  ARS,  we  have  the  declination  RS,^,\>  * 
and  the  angle  A  to  find  AR.     Therefore,  ,   £i>&  (*&  : 

Rad.  xsin.  AR^cot.  A  xtan.  RS. 

from  the  middle  part  by  the  angles  A  and  B.   Hence,  rad.  cos.  c=cos.  a  cos.  b  .•.  cos.  c= 
cos.,  cos.  6 
rad. 

2.  In  the  spherical  triangle,  right  angled  at  C,  are  given  two  perpendicular  sides, 
viz.  a=116°  30'  43",  i=29°  41' 32",  to  find  the  angle  A. 

Here,  the  required  angle  is  adjacent  to  one  of  the  given  parts,  viz.  6,  which  make 
the  middle  part.    Then, 

Rad.xsin,  6=cot  A  tan.  a  .-.cot.  A=rad-Xsin'  6=76o  7/U/r. 

tan.  a. 


64 


THE  EARTH. 


Ex.    1.    Required  the  time  of  sunrise  at  latitude  52°  13'  N 
when  the  sun's  declination  is  23°  28'. 

Rad 10. 

•*  10.1105786 
L  9.6376106 


Cot.  A  or  tan.  52°  13' 
Tan.  BS=       23°  28' 

Sin.  34°  03  21i" 

4* 


9.7481892 


2h.  16'  13"  25'"  ; 

6 

3h.  43' 46"  35"'= the   time   after    midnight,   and  of 
course  the  time  of  rising. 

Ex.  2.  Required  the  time  of  sunrise  at  latitude  57°  2'  54"  N. 
when  the  sun's  declination  is  23°  28'  N. 

Ans.  3h.  llm.  49s. 

Ex.  3.  How  long  is  the  sun  above  the  horizon  in  latitude  58° 
12'  N.  when  his  declination  is  18°  40'  S.  ? 

Ans.  7h.  35m.  52s. 

134.  Given  the  Latitude  of  the  place,  and  the  Declination  of  a 
heavenly  body,  to  determine  its  Altitude  and  Azimuth  when  on  the 
six  o'clock  hour  circle. 

Let  HZO  (Fig.  23,)  be  the  meridian  of  the  place,  Z  the  zenith. 


Fig.  23. 


HO  the  horizon,  S  the  place  of 
the  object  on  the  6  o'clock  hour 
circle  PSP',  which  of  course  cuts 
the  equator  in  the  east  and  west 
points,  and  ZSB  the  vertical  cir- 
cle passing  through  the  body. 
Then  in  the  right,angled  triangle 
SBA,  the  given  quantities  are 
AS,  which  is  the  declination, 
and  the  arc  OP  or  angle  SAB, 
the  latitude  of  the  place,  to  find 
the  altitude  BS,  and  the  azimuth 
BO,  or  the  amplitude  AB,  which  is  its  complement. 

Ex.  1.  What  were  the  altitude  and  azimuth  of  Arcturus,  when 


Degrees  are  converted  into  hours  by  multiplying  by  4  and  dividing  by  60. 


ASTRONOMICAL    PROBLEMS. 


65 


upon  the  six  o'clock  hour  circle  of  Greenwich,  lat.  51°  28'  40"  N. 
on  the  first  of  April,  1822  ;  its  declination  being  20°  6'  50"  N.? 


For  the  Attitude. 

Rad.  sin.   BS=sin.  AS   sin.  A 
Rad.         .         .    .     10. 
Sin.  20°  06'  50"         9.5364162 
Sin.  51    28  40  9.8934103 


Sin.  15    36  27 


9.4298265 


For  the  Azimuth. 

Rad.  cos.  A=cot.  BO  cot.  AS 
Cot.  20°  06'  50"       10.4362545 
Cos.  51    28  40  9.7943612 

Rad.N     '.  10. 


Cot.  77°  09'  04' 


9.3581067 


Ex.  2.  At  latitude  62°  12'  N.  the  altitude  of  the  sun  at  6  o'clock 
in  the  morning  was  found  to  be  18°  20'  23":  required  his  declina- 
tion and  azimuth. 

Ans.  Dec.  20°  50'  12"  N.     Az.  79°  56'  4". 

135.  The  Latitudes  and  Longitudes  of  two  celestial  objects  be- 
ing given,  to  find  their  Distance  apart. 

Let  P  (Fig.  24,)  represent  the  pole  of  the  ecliptic,  and  PS,  PS', 
two  arcs  of  celestial  latitude  (Art.  37,)  drawn  to  the  two  objects 
SS' ;  then  will  these  arcs  represent  the  Fig.  24 

co-latitudes,  the  angle  P  will  be  the 
difference  of  longitude,  and  the  arc  SS' 
will  be  the  distance  sought.  Here  we 
have  the  two  sides  and  the  included 
angle  given  to  find  the  third  side.  By 
Napier's  Rules  for  the  solution  of  oblique  angled  spherical  triangles, 
(see  Spherical  Trigonometry,)  the  sum  and  difference  of  the  two 
angles  opposite  the  given  sides  may  be  found,  and  thence  the  an- 
gles themselves.  The  required  side  may  then  be  found  by  the  theo- 
rem, that  the  sines  of  the  sides  are  as  the  sines  of  their  opposite 
angles.*  The  computation  is  omitted  here  on  account  of  its  great 
length.  If  P  be  the  pole  of  the  equator  instead  of  the  ecliptic, 
then  PS  and  PS'  will  represent  arcs  of  co-declination,  and  the 
angle  P  will,  denote  difference  of  right  ascension.  From  these 
data,  also,  we  may  therefore  derive  the  distance  between  any  two 
stars.  Or,  finally,  if  P  be  the  pole  of  the  horizon,  the  angle  at  P 

*  More  concise  formulae  for  the  solution  of  this  case  may  be  found  in  Young's  Tri- 
gonometry, p.  99. — Francoeur's  Uranography,  Art.  330. — Dr.  Bowditch's  Practical 
Navigator,  p.  436. 


66  THE    EARTH. 

will  denote  difference  of  azimuth,  and  the  sides  PS,  PS',  zenith 
distances,  from  which  the  side  SS'  may  likewise  be  determined. 

FIGURE    AND    DENSITY    OF   THE    EARTH. 

136.  We  have  already  shown,  (Art.  8,)  that  the  figure  of  the 
earth  is  nearly  globular  ;  but  since  the  semi-diameter  of  the  earth 
is  taken  as  the  base  line  in  determining  the  parallax  of  the  heav- 
enly bodies,  and  lies  therefore  at  the  foundation  of  all  astronomi- 
cfrl  measurements,  it  is  very  important  that  it  should  be  ascertained 
with  the  greatest  possible  exactness.     Having  now  learned  the 
use  of  astronomical  instruments,  and  the  method  of  measuring 
arcs  on  the  celestial  sphere,  we  are  prepared  to  understand  the 
methods   employed  to  determine  the  exact  figure  of  the  earth. 
This  element  is  indeed,  ascertained  in  four  different  ways,  each 
of  which  is  independent  of  all  the  rest,  namely,  by  investigating 
the  effects  of  the  centrifugal  force  arising  from  the  revolution  of 
the  earth  on  its  axis — by  measuring  arcs  of  the  meridian — by 
experiments  with  the  pendulum — and  by  the  unequal  action  of  the 
earth  on  the  moon,  arising  from  the  redundance  of  matter  about 
the  equatorial  regions.     We  will  briefly  consider  each  of  these 
methods. 

137.  First,  the  known  effects  of  the  centrifugal  force,  would  give 
to  the  earth  a  spheroidal  figure,  elevated  in  the  equatorial,  and  flat- 
tened in  the  polar  regions. 

Had  the  earth  been  originally  constituted  (as  geologists  sup- 
pose) of  yielding  materials,  either  fluid  or  semi-fluid,  so  that 
its  particles  could  obey  their  mutual  attraction,  while  the  body 
remained  at  rest  it  would  spontaneously  assume  the  figure  of  a 
perfect  sphere  ;  as  soon,  however,  as  it  began  to  revolve  on  its 
axis,  the  greater  velocity  of  the  equatorial  regions  would  give  to 
them  a  greater  centrifugal  force,  and  cause  the  body  to  swell  out 
into  the  form  of  an  oblate  spheroid.*  Even  had  the  solid  part  of 
the  earth  consisted  of  unyielding  materials  and  been  created  a 
perfect  sphere,  still  the  waters  that  covered  it  would  have  receded 
from  the  polar  and  have  been  accumulated  in  the  equatorial  re- 

*  See  a  good  explanation  of  this  subject  in  the  Edinburgh  Encyclopaedia,  II.  665. 


FIGURE    OF   THE    EARTH. 


67 


gions,  leaving  bare  extensive  regions  on  the  one  side,  and  ascend- 
ing to  a  mountainous  elevation  on  the  other. 

On  estimating  from  the  known  dimensions  of  the  earth  and 
the  velocity  of  its  rotation,  the  amount  of  the  centrifugal  force  in 
different  latitudes,  and  the  figure  of  equilibrium  which  would 
result,  Newton  inferred  that  the  earth  must  have  the  form  of  an 
oblate  spheroid  before  the  fact  had  been  established  by  observa- 
tion ;  and  he  assigned  nearly  the  true  ratio  of  the  polar  and  equa- 
torial diameters. 

138.  Secondly,  the  spheroidal  figure  of  the  earth  is  proved,  by 
actually  measuring  the  length  of  a  degree  on  the  meridian  in  differ- 
ent latitudes. 

Were  the  earth  a  perfect  sphere,  the  section  of  it  made  by  a 
plane  passing  through  its  center  in  any  direction  would  be  a  per- 
fect circle,  whose  curvature  would  be  equal  in  all  parts ;  but  if 
we  find  by  actual  observation,  that  the  curvature  of  the  section  is 
not  uniform,  we  infer  a  corresponding  departure  in  the  earth  from 
the  figure  of  a  perfect  sphere.  This  task  of  measuring  portions  of 
the  meridian,  has  been  executed  in  different  countries  by  means 
of  a  system  of  triangles  with  astonishing  accuracy.*  The  result 
is,  that  the  length  of  a  degree  increases  as  we  proceed  from  the 
equator  towards  the  pole,  as  may  be  seen  from  the  following  table : 


Places  of  observation. 

Latitude. 

Length  of  a  degree  in  miles. 

Peru, 
Pennsylvania, 
Italy, 
France, 
England, 
Sweden, 

00°  00'  00" 
39     12    00 
43     01    00 
46     12    00 
51     29    54| 
66    20    10 

68.732 
68.896 
68.998 
69.054 
69.146 
69.292 

Combining  the  results  of  various  measurements,  the  dimensions 
of  the  terrestrial  spheroid  are  found  to  be  as  follows  :  f 

Equatorial  diameter,  .         .         .         7925.308 

Polar  diameter,  ....         7898.952 

Mean  diameter,          ....         7912.130 

The  difference  between  the  greatest  and  least,  is  26.356=^ 


*  See  Day's  Trigonometry. 


t  Bessel. 


68  THE    EARTH. 

of  the  greatest.  This  fraction  (^IT)  is  denominated  the  elliplicity 
of  the  earth,  being  the  excess  of  the  transverse  over  the  conjugate 
axis,  on  the  supposition  that  the  section  of  the  earth  coinciding 
with  the  meridian,  is  an  ellipse  :  and  that  such  is  the  case,  is 
proved  by  the  fact  that  calculations  on  this  hypothesis,  of  the 
lengths  of  arcs  of  the  meridian  in  different  latitudes,  agree  nearly 
with  the  lengths  obtained  by  actual  measurement. 

139.  Thirdly,  the  figure  of  the  earth  is  shown  to  be  spheroidal,  by 
observations  with  the  pendulum. 

The  use  of  the  pendulum  in  determining  the  figure  of  the 
earth,  is  founded  upon  the  principle  that  the  number  of  vibra- 
tions performed  by  the  same  pendulum,  when  acted  on  by  differ 
ent  forces,  varies  as  the  square  root  of  the  forces.*  Hence,  by 
carrying  a  pendulum  to  different  parts  of  the  earth,  and  counting 
the  number  of  vibrations  it  performs  in  a  given  time,  we  obtain 
the  relative  forces  of  gravity  at  those  places,  and  this  leads  to  a 
knowledge  of  the  relative  distance  of  each  place  from  the  center 
of  the  earth,  and  finally,  to  the  ratio  between  the  equatorial  and 
the  polar  diameters. 

140.  Fourthly,  that  the  earth  is  of  a  spheroidal  figure,  is  infer- 
red from  tJie  motions  of  the  moon. 

These  are  found  to  be  affected  by  the  excess  of  matter  about 
the  equatorial  regions,  producing  certain  irregularities  in  the  lunar 
motions,  the  amount  of  which  becomes  a  measure  of  the  excess 
itself,  and  hence  affords  the  means  of  determining  the  earth's 
ellipticity.  This  calculation  has  been  made  by  the  most  profound 
mathematicians,  and  the  figure  deduced  from  this  source  corres- 
ponds very  nearly  to  that  derived  from  the  several  other  indepen 
dent  methods. 

We  thus  have  the  shape  of  the  earth  established  upon  the  most 
satisfactory  evidence,  and  are  furnished  with  a  starting  point  from 
which  to  determine  various  measurements  among  the  heavenly 
bodies.  x 

141.  The  density  of  the  earth  compared  with  water,  that  is,  its 

*  Mechanics,  Art.  183. 


DENSITY   OF  THE  EARTH. 

specific  gravity,  is  5£.*  The  density  was  first  estimated  by  Dr 
Hutton,  from  observations  made  by  Dr.  Maskelyne,  Astronomer 
Royal,  on  Schehallien,  a  mountain  of  Scotland,  in  the  year  1774. 
Thus,  let  M  (Fig.  25,)  represent  Fig.  25. 

the  mountain,  D,  B,  two  stations 
on  opposite  sides  of  the  moun- 
tain, and  I  a  star;  and  let  IE 
and  IG  be  the  zenith  distances  as 
determined  by  the  differences  of 
latitudes  of  the  two  stations.  But 
the  apparent  zenith  distances  as 
determined  by  the  plumb  line 
are  IE'  and  IG'.  The  deviation 
towards  the  mountain  on  each 
side  exceeded  7".f  The  attrac- 
tion of  the  mountain  being  ob- 
served on  both  sides  of  it,  and 
its  mass  being  computed  from  a  number  of  sections  taken  in  all 
directions,  these  data,  when  compared  with  the  known  attraction 
and  magnitude  of  the  earth,  led  to  a  knowledge  of  its  mean  den- 
sity. According  to  Dr.  Hutton,  this  is  to  that  of  water  as  9  to  2  ; 
but  later  and  more  accurate  estimates  have  made  the  specific 
gravity  of  the  earth  as  stated  above.  But  this  density  is  nearly 
double  the  average  density  of  the  materials  that  compose  the  ex- 
terior crust  of  the  earth,  showing  a  great  increase  of  density 
towards  the  center. 

The  density  of  the  earth  is  an  important  element,  as  we  shall 
find  that  it  helps  us  to  a  knowledge  of  the  density  of  each  of  the 
other  members  of  the  solar  system. 


»  Daily,  Ast  Tables,  p.  21. 


t  Robison's  Phys.  Ast. 


PART  II. — OF  THE  SOLAR  SYSTEM. 


142.  HAVING  considered  the  Earth,  in  its  astronomical  relations, 
and  the  Doctrine  of  the  Sphere,  we  proceed  now  to  a  survey  of 
the  Solar  System,  and  shall  treat  successively  of  the  Sun,  Moon, 
Planets,  and  Comets. 


CHAPTER  I. 

OF   THE    SUN SOLAR    SPOTS ZODIACAL    LIGHT. 

143.  THE  figure  which  the  sun  presents  to  us  is  that  of  a  per- 
fect circle,  whereas  most  of  the  planets  exhibit  a  disk  more  or  less 
elliptical,  indicating  that  the  true  shape  of  the  body  is  an  oblate 
spheroid.    So  great,  however,  is  the  distance  of  the  sun,  that  a 
line  400  miles  long  would  subtend  an  angle  of  only  1"  at  the  eye, 
and  would  therefore  be  the  least  space  that  could  be  measured. 
Hence,  were  the  difference  between  two  conjugate  diameters  of 
the  sun  any  quantity  less  than  this,  we  could  not  determine  by 
actual  measurement  that  it  existed  at  all.     Still  we  learn  from 
theoretical  considerations,  founded  upon  the  known  effects  of  cen- 
trifugal force,  arising  from  the  sun's  revolution  on  his  axis,  that 
his  figure  is  not  a  perfect  sphere,  but  is  slightly  spheroidal.* 

144.  The  distance  of  the  sun  from  the  earth,  is  nearly  95,000,000 
miles.     For,  its  horizontal  parallax  being  8."6,  (Art.  86,)  and  the 
semi-diameter  of  the  earth  3956  miles, 

Sin.  8."6  :  3956  : :  Rad.  :  95,000,000  nearly.  In  order  to  form 
some  faint  conception  at  least  of  this  vast  distance,  let  us  reflect 
that  a  railway  car,  moving  at  the  rate  of  20  miles  per  hour,  would 
require  more  than  500  years  to  reach  the  sun. 

*  See  Mecanique  Celeste,  III,  165.    Delambre,  t,  I,  p.  48a 


SOLAR  SPOTS.  71 

145.  The  apparent  diameter  of  the  sun  may  be  found  either  by 
the  Sextant,  (Art.  129,)  by  an  instrument  called  the  Heliometer, 
specially  designed  for  measuring  its  angular  breadth,  or  by  the  time 
it  occupies  in  crossing  the  meridian.  If,  for  example,  it  occupied 
4m,  its  angular  diameter  would  be  1°.  It  in  fact  occupies  a  little 
more  than  2m,  and  hence  its  apparent  diameter  is  a  little  more  than 
half  a  degree,  (32'  3").  Having  the  distance  and  angular  diameter, 
we  can  easily  find  its  linear  diameter.  Let  E  (Fig.  26,)  be  the 
earth,  S  the  sun,  ES  a  line  drawn  to  the  Fig.  26. 

center  of  the  disk,  and  EC  a  line  drawn 
touching  the  disk  at  C.     Join  SC  ;  then 

Rad.  :  ES  (95,000,000)  : :  sin.  16'  l."5 : 
442840=semi-diameter,  and  885680=diam-         \  ^fc 


eter.     And— -  —=112  nearly ;  that  is,  it 


would  require  one  hundred  and  twelve  bo- 
dies like  the  earth,  if  laid  side  by  side,  to 
reach  across  the  diameter  of  the  sun;  and  a 
ship  sailing  at  the  rate  of  ten  knots  an  hour, 
would  require  more  than  ten  years  to  sail 
across  the  solar  disk.  Since  spheres  are  to 
each  other  as  the  cubes  of  their  diameters, 

I3  :  1123  :  :  1  :  1,400,000  nearly;  that  is,  the  sun  is  about 
1,400,000  times  as  large  as  the  earth.  The  distance  of  the  moon 
from  the  earth  being  237,000  miles,  were  the  center  of  the  sun 
made  to  coincide  with  the  center  of  the  earth,  the  sun  would  ex- 
tend every  way  from  the  earth  nearly  twice  as  far  as  the  moon. 

146.  In  density,  the  sun  is  only  one  fourth  that  of  the  earth, 
being  but  a  little  heavier  than  water  (Art.  141)  ;  and  since  the 
quantity  of  matter,  or  mass  of  a  body,  is  proportioned  to  its  mag- 
nitude and  density,  hence,  1,400,000  x  £  =  350,000,  that  is,  the 
quantity  of  matter  in  the  sun  is  three  hundred  and  fifty  thousand 
(or,  more  accurately,  354,936)  times  as  great  as  in  the  earth.  Now 
the  weight  of  bodies  (which  is  a  measure  of  the  force  of  gravity) 
varies  directly  as  the  quantity  of  matter,  and  inversely  as  the 
square  of  the  distance.  A  body,  therefore,  would  weigh  350,000 
times  as  much  on  the  surface  of  the  sun  as  on  the  earth,  if  the 


72  THE   SUN. 

distance  of  the  center  of  force  were  the  same  in  both  cases  ;  but 
since  the  attraction  of  a  sphere  is  the  same  as  though  all  the  mat- 
ter were  collected  in  the  center,  consequently,  the  weight  of  a 
body,  so  far  as  it  depends  on  its  distance  from  the  center  of  force, 
would  be  the  square  of  112  times  less  at  the  sun  than  at  the  earth. 
Or,  putting  W  for  the  weight  at  the  earth,  and  W  for  the  weight 
at  the  sun,  then 


Hence  a  body  would  weigh  nearly  28  times  as  much  at  the  sun 
as  at  the  earth.  A  man  weighing  200  Ibs.  would,  if  transported 
to  the  surface  of  the  sun,  weigh  5,580  Ibs.,  or  nearly  2i  tons.  To 
lift  one's  limbs,  would,  in  such  a  case,  be  beyond  the  ordinary 
power  of  the  muscles.  At  the  surface  of  the  earth,  a  body  falls 
through  IGjL  feet  in  a  second  ;  and  since  the  spaces  are  as  the 
velocities,  the  times  being  equal,  and  the  velocities  as  the  forces, 
therefore  a  body  would  fall  at  the  sun  in  one  second,  through 
16TV  x  27T9o  =  448.7  feet. 


SOLAR  SPOTS. 


147.  The  surface  of  the  sun,  when  viewed  with  a  telescope, 
usually  exhibits  dark  spots,  which  vary  much,  at  different  times, 
in  number,  figure,  and  extent.  One  hundred  or  more,  assembled 
in  several  distinct  groups,  are  sometimes  visible  at  once  on  the 
solar  disk.  The  solar  spots  are  commonly  very  small,  but 
occasionally  a  spot  of  enormous  size  is  seen  occupying  an  extent 
of  50,000  miles  or  more  in  diameter.  They  are  sometimes 
even  visible  to  the  naked  eye,  when  the  sun  is  viewed  through 
colored  glass,  or  when  near  the  horizon,  it  is  seen  through  light 
clouds  or  vapors.  When  it  is  recollected  that  1"  of  the  solar 
disk  implies  an  extent  of  400  miles,  (Art.  143,)  it  is  evident  that  a 
space  large  enough  to  be  seen  by  the  naked  eye,  must  cover  a  very 
large  extent. 

A  solar  spot  usually  consists  of  two  parts,  the  nucleus  and  the 
umbra,  (Fig.  27.)  The  nucleus  is  black,  of  a  very  irregular  shape, 
and  is  subject  to  great  and  sudden  changes,  both  in  form  and  size. 
Spots  have  sometimes  seemed  to  burst  asunder,  and  to  project  frag- 
ments in  different  directions.  The  umbra  is  a  wide  margin  of  lighter 


SOLAR   SPOTS. 


shade,  and  is  commonly  of  greater  Fi£-  27- 

extent  than  the  nucleus.  The  spots 
are  usually  confined  to  a  zone  ex- 
tending across  the  central  regions 
of  the  sun,  riot  exceeding  60°  in 
breadth.  When  the  spots  are  ob- 
served from  day  to  day,  they  are 
seen  to  move  across  the  disk  of  the 
sun,  occupying  about  two  weeks  in 
passing  from  one  limb  to  the  other. 
After  an  absence  of  about  the  same 
period,  the  spot  returns,  having  taken  27d.  7h.  37m.  in  the  entire 
revolution. 

148.  The  spots  must  be  nearly  or  quite  in  contact  with  the  body 
of  the  sun.  Were  they  at  any  considerable  distance  from  it,  the 
time  during  which  they  would  be  seen  on  the  solar  disk,  would 
be  less  than  that  occupied  in  the  remainder  of  the  revolution. 
Thus,  let  S  (Fig.  28,)  be  the  sun,  E  the  earth,  and  abc  the  path 
of  the  body,  revolving  about  the  sun. 
Unless  the  spot  were  nearly  or  quite 
in  contact  with  the  body  of  the  sun, 
being  projected  upon  his  disk  only 
while  passing  from  b  to  c,  and  being 
invisible  while  describing  the  arc  cab, 
it  would  of  course  be  out  of  sight  lon- 
ger, than  in  sight,  whereas  the  two  pe- 
riods are  found  to  be  equal.  Moreover, 
the  lines  which  all  the  solar  spots  de- 
scribe on  the  disk  of  the  sun,  are  found 
to  be  parallel  to  each  other,  like  the 
circles  of  diurnal  revolution  around  the 
earth ;  and  hence  it  is  inferred  that 
they  arise  from  a  similar  cause,  namely, 
the  revolution  of  the  sun  on  his  aocis, 
a  fact  which  is  thus  made  known  to 


us. 


But  although  the  spots  occupy  about  27£  days  in  passing  from 

10 


74  THE  SUN. 

one  limb  of  the  sun  around  to  the  same  limb  again,  yet  this  is  not 

the  period  of  the  sun's  revolution  on  his  axis,  but  exceeds  it  by 

nearly  two  days.     For,  let  AA'B  (Fig.  29,)  represent  the  sun,  and 

EE'M   the  orbit   of  the   earth.     When  the   earth   is  at  E,  the 

visible  disk  of  the  sun  will  be  AA'B ; 

and  if  the  earth  remained  stationary  at 

E,  the  time  occupied  by  a  spot  after 

leaving  A  until  it  returned  to  A,  would 

be  just  equal  to  the  time  of  the  sun's 

revolution  on  his  axis.     But  during  the 

27|  days  in  which  the  spot  has  been 

performing  its  apparent  revolution,  the 

earth  has  been  advancing  in  his  orbit 

from  E  to  E',  where  the  visble  disk  of 

the  sun  is  A'B'.     Consequently,  before 

the  spot  can  appear  again  on  the  limb  from  which  it  set  out,  it 

must  describe  so  much  more  than  an  entire  revolution  as  equals 

the  arc  AA',  which  equals  the  arc  EE'.     Hence, 

365d.  5h.  48m.+27d.  7h.  37m. :  365d.  5h.  48m. : :  27d.  7h.  37m. : 
25d.  9h.  59m.=the  time  of  the  sun's  revolution  on  his  axis. 

149.  If  the  path  which  the  spots  appear  to  describe  by  the  re- 
volution of  the  sun  on  his  axis  left  each  a  visible  trace  on  his  sur- 
face, they  would  form,  like  the  circles  of  diurnal  revolution  on  the 
earth,  so  many  parallel  rings,  of  which  that  which  passed  through 
the  center  would  constitute  the  solar  equator,  while  those  on  each 
side  of  this  great  circle  would  be  small  circles,  corresponding  to 
parallels  of  latitude  on  the  earth.  Let  us  conceive  of  an  artifi- 
cial sphere  to  represent  the  sun,  having  such  rings  plainly  marked 
on  its  surface.  Let  this  sphere  be  placed  at  some  distance  from 
the  eye,  with  its  axis  perpendicular  to  the  axis  of  vision,  in  which 
case  the  equator  would  coincide  with  the  line  of  vision,  and  its 
edge  be  presented  to  the  eye.  It  would  therefore  be  projected  in- 
to a  straight  line.  The  same  would  be  the  case  with  all  the  small- 
er rings,  the  distance  being  supposed  such  that  the  rays  of  light 
come  from  them  all  to  the  eye  nearly  parallel.  Now  let  the  axis, 
instead  of  being  perpendicular  to  the  line  of  vision,  be  inclined  to 
that  line,  then  all  the  rings  being  seen  obliquely  would  be  projected 


« 


SOLAR    SPOTS.  75 

into  ellipses.  If,  however,  while  the  sphere  remained  in  a  fixed 
position,  the  eye  were  carried  around  it,  (being  always  in  the  same 
plane,)  twice  during  the  circuit  it  would  be  in  the  plane  of  the 
equator,  and  project  this  and  all  the  smaller  circles  into  straight 
lines ;  and  twice,  at  points  90°  distant  from  the  foregoing  posi- 
tions, the  eye  would  be  at  a  distance  from  the  planes  of  the  rings 
equal  to  the  inclination  of  the  equator  of  the  sphere  to  the  line  of 
vision.  Here  it  would  project  the  rings  into  wider  ellipses  than 
at  other  points  ;  and  the  ellipses  would  become  more  and  more 
acute  as  the  eye  departed  from  either  of  these  points,  until  they 
vanished  again  into  straight  lines. 

150.  It  is  in  a  similar  manner  that  the  eye  views  the  paths  de- 
scribed by  the  spots  on  the  sun.  If  the  sun  revolved  on  an  axis 
perpendicular  to  the  plane  of  the  earth's  orbit,  the  eye  being  situ- 
ated in  the  plane  of  revolution,  and  at  such  a  distance  from  the 
sun  that  the  light  comes  to  the  eye  from  all  parts  of  the  solar 
disk  nearly  parallel,  the  paths  described  by  the  spots  would  be 
projected  into  straight  lines,  and  each  would  describe  a  straight 
line  across  the  solar  disk,  parallel  to  the  plane  of  revolution.  But 
the  axis  of  the  sun  is  inclined  to  the  ecliptic  about  7£°  from  a  per- 
pendicular, so  that  usually  all  the  circles  described  by  the  spots  are 
projected  into  ellipses.  The  breadth  of  these,  however,  will  vary 
as  the  eye,  in  the  annual  revolution,  is  carried  around  the  sun,  and 
when  the  eye  comes  into  the  plane  of  the  rings,  as  it  does  twice  a 
year,  they  are  projected  into  straight  lines,  and  for  a  short  time  a 
spot  seems  moving  in  a  straight  line  inclined  to  the  plane  of  the 
ecliptic  7£°.  The  two  points  where  the  sun's  equator  cuts  the 
ecliptic  are  called  the  sun's  nodes.  The  longitudes  of  the  nodes 
are  80°  7'  and  260°  7',  and  the  earth  passes  through  them  about 
the  12th  of  December,  and  the  llth  of  June.  It  is  at  these  times 
that  the  spots  appear  to  describe  straight  lines.  We  have  men- 
tioned the  various  changes  in  the  apparent  paths  of  the  solar  spots, 
which  arise  from  the  inclination  of  the  sun's  axis  to  the  plane  of 
the  ecliptic ;  but  it  was  in  fact  by  first  observing  these  changes, 
and  proceeding  in  the  reverse  order  from  that  which  we  have  pur- 
sued, that  astronomers  ascertained  that  the  sun  revolves  on  his 
axis,  and  that  this  axis  is  inclined  to  the  ecliptic  82f°. 


76 


THE  SUN. 


151.  With  regard  to  the  cause  of  the  solar  spots,  various  hypo- 
theses have  been  proposed,  none  of  which  is  entirely  satisfactory. 
That  which  ascribes  their  origin  to  volcanic  action,  appears  to  us 
the  most  reasonable.* 

Besides  the  dark  spots  on  the  sun,  there  are  also  seen,  in  dif- 
ferent parts,  places  that  are  brighter  than  the  neighboring  por- 
tions of  the  disk.  These  are  called  faculce.  Other  inequalities 
are  observable  in  powerful  telescopes,  all  indicating  that  the  sur- 
face of  the  sun  is  in  a  state  of  constant  and  powerful  agitation. 

ZODIACAL    LIGHT. 

152.  The  Zodiacal  Light  is  a  faint  light  resembling  the  tail  of 
a  comet,  and  is  seen  at  certain  seasons  of  the  year  following  the 
course  of  the  sun  after  evening  twilight,  or  preceding  his  approach 
in  the  morning  sky.     Figure  30  represents  its  appearance  as  seen 
in  the  evening  in  March,  1836.     The  following  are  the  leading 
facts  respecting  it. 

1 .  Its  form  is  that  of  a  luminous  Fig.  30. 
pyramid,  having   its   base    towards 

the  sun.  It  reaches  to  an  immense 
distance  from  the  sun,  sometimes 
even  beyond  the  orbit  of  the  earth. 
It  is  brighter  in  the  parts  nearer  the 
sun  than  in  those  that  are  more 
remote,  and  terminates  in  an  ob- 
tuse apex,  its  light  fading  away  by 
insensible  gradations,  until  it  be- 
comes too  feeble  for  distinct  vision. 
Hence  its  limits  are,  at  the  same 
time,  fixed  •  at  different  distances 
from  the  sun  by  different  observers, 
according  to  their  respective  powers 
of  vision. 

2.  Its  aspects  vary  very  much  with  the  different  seasons  of  the 
year.     About  the  first  of  October,  in  our  climate,  (Lat.  41°  18',) 

*  In  the  system  of  instruction  in  Yale  College,  subjects  of  this  kind  are  discussed 
in  a  course  of  astronomical  lectures,  addressed  to  the  class  after  they  have  finished  the 
perusal  of  the  text-book. 


ZODIACAL    LIGHT,  77 

it  becomes  visible  before  the  dawn  of  day,  rising  along  north  of 
the  ecliptic,  and  terminating  above  the  nebula  of  Cancer.  About 
the  middle  of  November,  its  vertex  is  in  the  constellation  Leo. 
At  this  time  no  traces  of  it  are  seen  in  the  west  after  sunset,  but 
about  the  first  of  December  it  becomes  faintly  visible  in  the  west, 
crossing  the  Milky  Way  near  the  horizon,  and  reaching  from  the 
sun  to  the  head  of  Capricornus,  forming,  as  its  brightness  increases, 
a  counterpart  to  the  Milky  Way,  between  which  on  the  right, 
and  the  Zodiacal  Light  on  the  left,  lies  a  triangular  space  embra- 
cing the  Dolphin.  Through  the  month  of  December,  the  Zodi- 
acal Light  is  seen  on  both  sides  of  the  sun,  namely,  before  the 
morning  and  after  the  evening  twilight,  sometimes  extending  50° 
westward,  and  70°  eastward  of  the  sun  at  the  same  time.  After 
it  begins  to  appear  in  the  western  sky,  it  increases  rapidly  from 
night  to  night,  both  in  length  and  brightness,  and  withdraws  itself 
from  the  morning  sky,  where  it  is  scarcely  seen  after  the  month 
of  December,  until  the  next  October. 

3.  The  Zodiacal  Light  moves  through  the  heavens  in  the  order  of 
the  signs.     It  moves  with  unequal  velocity,  being  sometimes  sta- 
tionary and  sometimes  retrograde,  while  at  other  times  it  ad- 
vances much  faster  than  the  sun.     In  February  and  March,  it  is 
very  conspicuous  in  the  west,  reaching  to  the  Pleiades  and  be- 
yond ;  but  in  April  it  becomes  more  faint,  and  nearly  or  quite  dis- 
appears during  the  month  of  May.     It  is  scarcely  seen  in  this  lat- 
itude during  the  summer  months. 

4.  It  is  remarkably  conspicuous   at   certain  periods  of  a  few 
years,  and  then  far  a  long  interval  almost  disappears. 

5.  The  Zodiacal  Light  was  formerly  field  to  be  the  atmosphere  of 
the  sun.*     But  La  Place  has  shown  that  the  solar  atmosphere 
could  never  reach  so  far  from  the  sun  as  this  light  is  seen  to  ex- 
tend.f     It  has  been  supposed  by  others  to  be  a  nebulous  body 
revolving  around  the  sun.     The  idea  has  been  suggested,  that  the 
extraordinary  Meteoric  Showers,  which  at  different  periods  visit 
the  earth,  especially  in  the  month  of  November,  may  be  derived 
from  this  body.J 

*  Mairan,  Memoirs  French  Academy,  for  1733.  t  Mec.  Celeste,  III,  525. 

t  See  note  on  "  Meteoric  Showers,"  at  the  end  of  the  volume. 


CHAPTER    II. 

OP  THE  APPARENT  ANNUAL  MOTION  OF  THE  SUN SEASONS FIGURE 

OF  THE  EARTH'S  ORBIT. 

153.  THE  revolution  of  the  earth  around  the  sun  once  a  year, 
produces  an  apparent  motion  of  the  sun  around  the  earth  in  the 
same  period.  When  bodies  are  at  such  a  distance  from  each 
other  as  the  earth  and  the  sun,  a  spectator  on  either  would  pro- 
ject the  other  body  upon  the  concave  sphere  of  the  heavens,  al- 
ways seeing  it  on  the  opposite  side  of  a  great  circle,  180°  from 
himself.  Thus  when  the  earth  arrives  at  Libra  (Fig.  11,)  we  see 
the  sun  in  the  opposite  sign  Aries.  When  the  earth  moves  from 
Libra  to  Scorpio,  as  we  are  unconscious  of  our  own  motion,  the 
sun  it  is  that  appears  to  move  from  Aries  to  Taurus,  being  always 
seen  in  the  heavens,  where  a  line  drawn  from  the  eye  of  the  spec- 
tator through  the  body  meets  the  concave  sphere  of  the  heavens. 
Hence  the  line  of  projection  carries  the  sun  forward  on  one  side 
of  the  ecliptic,  at  the  same  rate  as  the  earth  moves  on  the  oppo- 
site side ;  and  therefore,  although  we  are  unconscious  of  our  own 
motion,  we  can  read  it  from  day  to  day  in  the  motions  of  the  sun. 
If  we  could  see  the  stars  at  the  same  time  with  the  sun,  we  could 
actually  observe  from  day  to  day  the  sun's  progress  through  them, 
as  we  observe  the  progress  of  the  moon  at  night ;  only  the  sun's 
rate  of  motion  would  be  nearly  fourteen  times  slower  than  that 
of  the  moon.  Although  we  do  not  see  the  stars  when  the  sun  is 
present,  yet  after  the  sun  is  set,  we  can  observe  that  it  makes  daily 
progress  eastward,  as  is  apparent  from  the  constellations  of  the 
Zodiac  occupying,  successively,  the  western  sky  after  sunset, 
proving  that  either  all  the  stars  have  a  common  motion  westward 
independent  of  their  diurnal  motion,  or  that  the  sun  has  a  motion 
past  them,  from  west  to  east.  We  shall  see  hereafter  abundant 
evidence  to  prove,  that  this  change  in  the  relative  position  of  the 
sun  and  stars,  is  owing  to  a  change  in  the  apparent  place  of  the 
sun,  and  not  to  any  change  in  the  stars. 


ANNUAL  MOTION.  79 

154.  Although  the  apparent  revolution  of  the  sun  is  in  a  direc- 
tion opposite  to  the  real  motion  of  the  earth,  as  regards  absolute 
space,  yet  both  are  nevertheless  from  west  to  east,  since  these 
terms  do  not  refer  to  any  directions  in  absolute  space,  but  to  the 
order  in  which  certain  constellations  (the  constellations  of  the 
Zodiac)  succeed  one  another.     The  earth  itself,  on  opposite  sides 
of  its  orbit,  does  in  fact  move  towards  directly  opposite  points  of 
space  ;  but  it  is  all  the  while  pursuing  its  course  in  the  order  of 
the  signs.     In  the  same  manner,  although  the  earth  turns  on  its 
axis  from  west  to  east,  yet  any  place  on  the  surface  of  the  earth 
is  moving  in  a  direction  in  space  exactly  opposite  to  its  direction 
twelve  hours  before.     If  the  sun  left  a  visible  trace  on  the  face 
of  the  sky,  the  ecliptic  would  of  course  be  distinctly  marked  on 
the  celestial  sphere  as  it  is  on  an  artificial  globe ;  and  were  the 
equator  delineated  in  a  similar  manner,  (by  any  method  like  that 
supposed  in  Art.  46,)  we  should  then  see  at  a  glance  the  relative 
position  of  these  two  circles,  the  points  where  they  intersect  one 
another  constituting  the  equinoxes,  the  points  where  they  are  at 
the  greatest  distance  asunder,  or  the  solstices,  and  various  other 
particulars,  which,  for  want  of  such  visible  traces,  we  are  now 
obliged  to  search  for  by  indirect  and  circuitous  methods.     It  will 
even  aid  the  learner  to  have  constantly  before  his  mental  vision, 
an  imaginary  delineation  of  these  two  important  circles  on  the 
face  of  the  sky. 

155.  The  method  of  ascertaining  the  nature  and  position  of  the 
earth's  orbit,  is  by  observations  on  the  sun's  Decimation  and  Right 
Ascension. 

The  exact  declination  of  the  sun  at  any  time  is  determined 
from  his  meridian  altitude  or  zenith  distance,  the  latitude  of  the 
place  of  observation  being  known,  (Art.  37.)  The  instant  the 
center  of  the  sun  is  on  the  meridian,  (which  instant  is  given  by 
the  transit  instrument,)  we  take  the  distance  of  his  upper  and 
that  of  his  lower  limb  from  the  zenith :  half  the  sum  of  the  two 
observations  corrected  for  refraction,  gives  the  zenith  distance  of 
the  center.  This  result  is  diminished  for  parallax,  (Art.  84,)  and 
we  obtain  the  zenith  distance  as  it  would  be  if  seen  from  the 
center  of  the  earth.  The  zenith  distance  being  known,  the  de- 


80  THE  SUN. 

ciination  is  readily  found,  by  subtracting  that  distance  from  the 
latitude.  By  thus  taking  the  sun's  declination  for  every  day  of 
the  year  at  noon,  and  comparing  the  results,  we  learn  its  motion 
to  and  from  the  equator. 

156.  To  obtain  the  motion  in  right  ascension,  we  observe,  with 
a  transit  instrument,  the  instant  when  the  center  of  the  sun  is  on 
the  meridian.  Our  sidereal  clock  gives  us  the  right  ascension  in 
time  (Art.  124,)  which  we  may  easily,  if  we  choose,  convert  into 
degrees  and  minutes,  although  it  is  more  common  to  express  right 
ascension  by  hours,  minutes,  and  seconds.  The  differences  of 
right  ascension  from  day  to  day  throughout  the  year,  give  us  the 
sun's  annual  motion  parallel  to  the  equator.  From  the  daily  re- 
cords of  these  two  motions,  at  right  angles  to  each  other,  arran- 
ged in  a  table,*  it  is  easy  to  trace  out  the  path  of  the  sun  on  the 
artificial  globe ;  or  to  calculate  it  with  the  greatest  precision  by 
means  of  spherical  triangles,  since  the  declination  and  right  ascen- 
sion constitute  two  sides  of  a  right  angled  spherical  triangle,  the 
corresponding  arc  of  the  ecliptic,  that  is,  the  longitude,  being  the 
third  side,  (Art.  132.)  By  inspecting  a  table  of  observations, 
we  shall  find  that  the  declination  attains  its  greatest  value  on 
the  22d  of  December,  when  it  is  23°  27'  54"  south ;  that  from 
this  period  it  diminishes  daily  and  becomes  nothing  on  the  21st 
of  March ;  that  it  then  increases  towards  the  north,  and  reaches 
a  similar  maximum  at  the  northern  tropic  about  the  22d  of  June ; 
and,  finally,  that  it  returns  again  to  the  southern  tropic  by  gra- 
dations similar  to  those  which  marked  its  northward  progress.  A 
table  of  observations  also  would  show  us,  that  the  daily  differences 
of  declination  are  very  unequal ;  that,  for  several  days,  when  the 
sun  is  near  either  tropic,  its  declination  scarcely  varies  at  all ; 
while  near  the  equator,  the  variations  from  day  to  day  are  very 
rapid, — a  fact  which  is  easily  understood,  when  we  reflect,  that 
at  the  solstices  the  equator  and  the  ecliptic  are  parallel  to  each 
other,f  both  being  at  right  angles  to  the  meridian ;  while  at  the 

*  Such  a  table  may  be  found  in  Blot's  Astronomy,  in  Delambre,  and  in  most  collec- 
tions of  Astronomical  Tables. 

t  Or,  more  properly,  the  tangents  of  the  two  circles  (which  denote  the  directions  of 
the  curves  at  those  points)  are  parallel. 


ANNUAL  MOTION.  81 

equinoxes,  the  ecliptic  departs  most  rapidly  from  the  direction  of 
the  equator. 

On  examining,  in  like  manner,  a  table  of  observations  of  the 
right  ascension,  we  find  that  the  daily  differences  of  right  ascen- 
sion are  likewise  unequal ;  that  the  mean  of  them  all  is  3m  56s, 
or  236s,  but  that  they  have  varied  between  215s  and  266s.  On 
examining,  moreover,  the  right  ascension  at  each  of  the  equi- 
noxes, we  find  that  the  two  records  differ  by  180°;  which  proves 
that  the  path  of  the  sun  is  a  great  circle,  since  no  other  would 
bisect  the  equinoctial  as  this  does. 

157.  The  obliquity  of  the  ecliptic  is  equal  to  the  sun's  greatest 
declination.     For,  by  article  22,  the  inclination  of  any  two  great 
circles  is  equal  to  their  greatest  distance  asunder,  as  measured  on 
the  sphere.     The  obliquity  of  the  ecliptic  may  be  determined 
from  the  sun's  meridian  altitude,  or  zenith  distance,  on  the  clay 
of  the  solstice.     The  exact  instant  of  the  solstice,  however,  will 
not  of  course  occur  when  the  sun  is  on  the  meridian,  but  may 
happen  at  some  other  meridian ;  still,  the  changes  of  declination 
near  the  solstice  are  so  exceedingly  small,  that  but  a  slight  error 
can  result  from  this  source.     The  obliquity  may  also  be  found, 
without  knowing  the  latitude,  by  observing  the  greatest  and  least 
meridian  altitudes  of  the  sun,  and  taking  half  the  difference. 
This  is  the  method  practiced  in  ancient  times  by  Hipparchus. 
(Art.  2.)     On  comparing  observations  made  at  different  periods 
for  more  than  two  thousand  years,  it  is  found,  that  the  obliquity 
of  the  ecliptic  is  not  constant,  but  that  it  undergoes  a  slight  dimi- 
nution from  age  to  age,  amounting  to  52"  in  a  century,  or  about 
half  a  second  annually.     We  might  apprehend  that  by  successive 
approaches  to  each  other  the  equator  and  ecliptic  would  finally 
coincide ;  but  astronomers  have  ascertained  by  an  investigation, 
founded  on  the  principles  of  universal  gravitation,  that  this  varia- 
tion is  confined  within  certain  narrow  limits,  and  that  the  obli- 
quity, after  diminishing  for  some  thousands  of  years,  will  then 
increase  for  a  similar  period,  and  will  thus  vibrate  for  ever  about 
a  mean  value. 

158.  The  dimensions  of  the  earth9 s  orbit,  when  compared  with  its 
own  magnitude,  are  immense. 

11 


85?  THE  SUN. 

Since  the  distance  of  the  earth  from  the  sun  is  95,000,000 
miles,  and  the  length  of  the  entire  orbit  nearly  600,000,000  miles, 
it  will  be  found,  on  calculation,  that  the  earth  moves  1,640,000 
miles  per  day,  68,000  miles  per  hour,  1,100  miles  per  minute,  and 
nearly  19  miles  every  second,  a  velocity  nearly  fifty  times  as  great 
as  the  maximum  velocity  of  a  cannon  ball.  A  place  on  the  earth's 
equator  turns,  in  the  diurnal  revolution,  at  the  rate  of  about  1,000 
miles  an  hour  and  T\  of  a  mile  per  second.  The  motion  around 
the  sun,  therefore,  is  nearly  70  times  as  swift  as  the  greatest  mo- 
tion around  the  axis. 

THE   SEASONS. 

159.  The  change  of  seasons  depends  on  two  causes,  (1)  the  ob- 
liquity of  the  ecliptic,  and  (2)  the  earth's  axis  always  remaining 
parallel  to  itself.     Had  the  earth's  axis  been  perpendicular  to  the 
plane  of  its  orbit,  the  equator  would  have  coincided  with  the 
ecliptic,  and  the  sun  would  have  constantly  appeared  in  the  equa- 
tor.    To  the  inhabitants  of  the  equatorial  regions,  the  sun  would 
always  have  appeared  to  move  in  the  prime  vertical ;  and  to  the 
inhabitants  of  either  pole,  he  would  always  have  been  in  the  ho- 
rizon.    But  the  axis  being  turned  out  of  a  perpendicular  direc- 
tion 23°  28',  the  equator  is  turned  the  same  distance  out  of  the 
ecliptic ;  and  since  the  equator  and  ecliptic  are  two  great  circles 
which  cut  each  other  in  two  opposite  points,  the  sun,  while  per- 
forming his  circuit  in  the  ecliptic,  must  evidently  be  once  a  year 
in  each  of  those  points,  and  must  depart  from  the  equator  of  the 
heavens  to  a  distance  on  either  side  equal  to  the  inclination  of  the 
two  circles,  that  is,  23°  28'.     (Art.  22.) 

160.  The  earth  being  a  globe,  the  sun  constantly  enlightens 
the  half  next  to  him,*  while  the  other  half  is  in  darkness.     The 
boundary  between  the  enlightened  and  the  unenlightened  part,  is 
called  the  circle  of  illumination.     When  the  earth  is  at  one  of 
the  equinoxes,  the  sun  is  at  the  other,  and  the  circle  of  illumina- 

*  In  fact,  the  sun  enlightens  a  little  more  than  half  the  earth,  since  on  account  of 
his  vast  magnitude  the  tangents  drawn  from  opposite  sides  of  the  sun  to  opposite  sides 
of  the  earth,  converge  to  a  point  behind  the  earth,  as  will  be  seen  by  and  by  in  the 
representation  of  eclipses.  The  amount  of  illumination  also  is  increased  by  refraction. 


THE    SEASONS. 

tion  passes  through  both  the  poles.  When  the  earth  reaches  one 
of  the  tropics,  the  sun  being  at  the  other,  the  circle  of  illumina- 
tion cuts  the  earth  so  as  to  pass  23°  28'  beyond  the  nearer,  and 
the  same  distance  short  of  the  remoter  pole.  These  results  would 
not  be  uniform,  were  not  the  earth's  axis  always  to  remain  parallel 
to  itself.  The  following  figure  will  illustrate  the  foregoing  state- 
ments. 

Fig.  31. 


Let  ABCD  represent  the  earth's  place  in  different  parts  of  its 
orbit,  having  the  sun  in  the  center.  Let  A,  C,  be  the  position  of 
the  earth  at  the  equinoxes,  and  B,  D,  its  positions  at  the  tropics, 
the  axis  ns  being  always  parallel  to  itself.*  At  A  and  C  the  sun 
shines  on  both  n  and  s ;  and  now  let  the  globe  be  turned  round 
on  its  axis,  and  the  learner  will  easily  conceive  that  the  sun  will 
appear  to  describe  the  equator,  which  being  bisected  by  the  hori- 


*  The  learner  will  remark  that  the  hemisphere  towards  n  is  above,  and  that  towards 
*  is  below  the  plane  of  the  paper.  It  is  important  to  form  a  just  conception  of  the 
position  of  the  axis  with  respect  to  the  plane  of  its  orbit. 


84  THE   SUN. 

zon  of  every  place,  of  course  the  day  and  night  will  be  equal  in  all 
parts  of  the  globe.*  Again,  at  B  when  the  earth  is  at  the  south- 
ern tropic,  the  sun  shines  23 1°  beyond  the  north  pole  n,  and  falls 
the  same  distance  short  of  the  south  pole  s.  The  case  is  exactly 
reversed  when  the  earth  is  at  the  northern  tropic  and  the  sun  at 
the  southern.  While  the  earth  is  at  one  of  the  tropics,  at  B  for 
example,  let  us  conceive  of  it  as  turning  on  its  axis,  and  we  shall 
readily  see  that  all  that  part  of  the  earth  which  lies  within  the 
north  polar  circle  will  enjoy  continual  day,  while  that  within  the 
south  polar  circle  will  have  continual  night,  and  that  all  other 
places  will  have  their  days  longer  as  they  are  nearer  to  the  en- 
lightened pole,  and  shorter  as  they  are  nearer  to  the  unenlightened 
pole.  This  figure  likewise  shows  the  successive  positions  of  the 
earth  at  different  periods  of  the  year,  with  respect  to  the  signs, 
and  what  months  correspond  to  particular  signs.  Thus  the  earth 
enters  Libra  and  the  sun  Aries  on  the  21st  of  March,  and  on  the 
21st  of  June  the  earth  is  just  entering  Capricorn  and  the  sun  Can- 
cer. 

161.  Had. the  axis  of  the  earth  been  perpendicular  to  the  plane 
of  the  ecliptic,  then  the  sun  would  always  have  appeared  to  move 
in  the  equator,  the  days  would  every  where  have  been  equal  to  the 
nights,  and  there  could  have  been  no  change  of  seasons.  On  the 
other  hand,  had  the  inclination  of  the  ecliptic  to  the  equator  been 
much  greater  than  it  is,  the  vicissitudes  of  the  seasons  would  have 
been  proportionally  greater  than  at  present.  Suppose,  for  instance, 
the  equator  had  been  at  right  angles  to  the  ecliptic,  in  which  case, 
the  poles  of  the  earth  would  have  been  situated  in  the  ecliptic 
itself;  then  in  different  parts  of  the  earth  the  appearances  would 
•  have  been  as  follows.  To  a  spectator  on  the  equator,  the  sun  as 
he  left  the  vernal  equinox  would  every  day  perform  his  diurnal 
revolution  in  a  smaller  and  smaller  circle,  until  he  reached  the 
north  pole,  when  he  would  halt  for  a  moment  and  then  wheel 
about  and  return  to  the  equator  in  the  reverse  order.  The  pro- 
gress of  the  sun  through  the  southern  signs,  to  the  south  pole, 
would  be  similar  to  that  already  described.  Such  would  be  the 

*  At  the  pole,  the  solar  disk,  at  the  time  of  tne  equinox,  appears  bisected  by  the  ho 


FIGURE   OP   THE   EARTIl's  ORBIT.  85 

appearances  to  an  inhabitant  of  the  equatorial  regions.  To  a 
spectator  living  in  an  oblique  sphere,  in  our  own  latitude  for  ex- 
ample, the  sun  while  north  of  the  equator  would  advance  continu- 
ally northward,  making  his  diurnal  circuits  in  parallels  further  and 
further  distant  from  the  equator,  until  he  reached  the  circle  of  per- 
petual apparition,  after  which  he  would  climb  by  a  spiral  course 
to  the  north  star,  and  then  as  rapidly  return  to  the  equator.  By  a 
similar  progress  southward,  the  sun  would  at  length  pass  the  circle 
of  perpetual  occultation,  and  for  some  time  (which  would  be 
longer  or  shorter  according  to  the  latitude  of  the  place  of  obser- 
vation) there  would  be  continual  night. 

The  great  vicissitudes  of  heat  and  cold  which  would  attend 
such  a  motion  of  the  sun,  would  be  wholly  incompatible  with  the 
existence  of  either  the  animal  or  the  vegetable  kingdoms,  and  all 
terrestrial  nature  would  be  doomed  to  perpetual  sterility  and  deso- 
lation. The  happy  provision  which  the  Creator  has  made  against 
such  extreme  vicissitudes,  by  confining  the  changes  of  the  seasons 
within  such  narrow  bounds,  conspires  with  many  other  express 
arrangements  in  the  economy  of  nature  to  secure  the  safety  and 
comfort  of  the  human  race. 


FIGURE   OF   THE    EARTH*S    ORBIT. 

162.  Thus  far  we  have  taken  the  earth's  orbit  as  a  great  circle, 
such  being  the  projection  of  it  on  the  celestial  sphere  ;  but  we  now 
proceed  to  investigate  its  actual  figure. 

Were  the  earth's  path  a  circle,  having  the  sun  in  the  center,  the 
sun  would  always  appear  to  be  at  the  same  distance  from  us ;  that 
is,  the  radius  of  its  orbit,  or  radius  vector,  the  name  given  to  a  line 
drawn  from  the  center  of  the  sun  to  the  orbit  of  any  planet, 
would  always  be  of  the  same  length.  But  the  earth's  distance 
from  the  sun  is  constantly  varying,  which  shows  that  its  orbit  is 
not  a  circle.  We  learn  the  true  figure  of  the  orbit,  by  ascertain- 
ing the  relative  distances  of  the  earth  from  the  sun  at  various  pe- 
riods of  the  year.  These  all  being  laid  down  in  a  diagram,  accord- 
ing to  their  respective  lengths,  the  extremities,  on  being  connected, 
give  us  our  first  idea  of  the  shape  of  the  orbit,  which  appears  of 
an  oval  form,  and  at  least  resembles  an  ellipse ;  and,  on  further 


86 


THE  SUN. 


trial,  we  find  that  it  has  the  properties  of  an  ellipse.  Thus,  let  E 
(Fig.  32,)  be  the  place  of  the  earth,  and  a,  bt  c,  &c.  successive  po- 
sitions of  the  sun ;  the  relative  lengths  of  the  lines  E#,  Eft,  &c.  be- 
ing known  on  connecting  the  points,  «,  b,  c,  &c.  the  resulting 
figure  indicates  the  true  shape  of  the  earth's  orbit. 

Fig.  32. 


163.  These  relative  distances  are  found  in  two  different  ways ; 
first,  by  changes  in  the  surfs  apparent  diameter,  and,  secondly,  by 
variations  in  his  angular  velocity.  Were  the  variations  in  the 
sun's  horizontal  parallax  considerable,  as  is  the  case  with  the 
moon's,  this  might  be  made  the  measure  of  the  relative  distances, 
for  the  parallax  varies  inversely  as  the  distance,  (Art.  82) ;  but  the 
whole  horizontal  parallax  of  the  sun  is  only  9",  and  its  variations 
are  too  slight  and  delicate,  and  too  difficult  to  be  found,  to  serve 
as  a  criterion  of  the  changes  in  the  sun's  distance  from  the  earth. 
But  the  changes  in  the  surfs  apparent  diameter,  are  much  more 
sensible,  and  furnish  a  better  method  of  measuring  the  relative 
distances  of  the  earth  from  the  sun.  By  a  principle  in  optics,  the 
apparent  diameter  of  an  object,  at  different  distances  from  the 
spectator,  is  inversely  as  the  distance.*  Hence,  the  apparent 
diameters  of  the  sun,  taken  at  different  periods  of  the  year,  be- 
come measures  of  the  different  lengths  of  the  radius  vector. 


*  More  exactly,  the  tangent  of  the  apparent  diameter  is  inversely  as  the  distance ; 
but  in  small  angles  like  those  concerned  in  the  present  inquiry,  the  angle  itself  may  be 
taken  for  the  tangent. 


87 

164.  The  point  where  the  earth,  or  any  planet,  in  its  revolution, 
is  nearest  the  sun,  is  called  its  perihelion :  the  point  where  it  is 
furthest  from  the  sun,  its  aphelion.     The  place  of  the  earth's  peri- 
helion is  known,  since  there  the  apparent  magnitude  of  the  sun  is 
greatest ;  and  when  the  sun's  magnitude  is  least,  the  earth  is 
known  to  be  at  its  aphelion.     The  sun's  apparent  diameter  when 
greatest  is  32'  35."6  ;  and  when  least,  31'  31";  hence  the  radius 
vector  at  the  aphelion  :  rad.  vector  at  the  perihelion  :  :  32.5933  : 
31.5167  ::  1.034  :  1.     Half  of  the  difference  of  the  two  is  equal 
to  the  distance  of  the  focus  of  the  ellipse  from  the  center,  a  quan- 
tity which  is  always  taken  as  the  measure  of  the  eccentricity  of  a 
planetary  orbit. 

165.  The  differences  of  angular  velocity  in  the  sun  in  the  dif- 
ferent parts  of  his  apparent  revolution,  are  still  more  remarkable. 
At  the  perihelion,  the  sun  moves  in  twenty-four  hours  over  an  arc 
of  61',  while  at  the  aphelion  he  describes  in  the  same  time  an  arc 
of  only  57',  these  being  the  daily  increments  of  longitude  in  those 
two  points  respectively.     If  the  apparent  motions  of  the  sun  de- 
pended alone  on  our  different  distances  from  him,  the  angular  ve- 
locity would  vary  inversely  as  the  distance,  and  the  ratio  expressed 
by  these  two  numbers  would  be  the  same  as  that  of  the  two  num- 
bers which  denote  the  differences  of  apparent  diameter  in  these 

fil  09  r:qqq 

two  points.     That  is,  2f  (=1.07)  would  equal  -       —  (  =  1.034)  ; 
«)•  ol.51oT 

but  the  first  fraction  is  equal  to  the  square  of  the  second,  for  1.07= 
1 .0342.  Hence,  the  surfs  angular  velocities  are  to  each  other  inversely 
as  the  squares  of  the  distances  at  the  perihelion  and  the  aphelion  ;  and 
by  a  similar  method,  the  same  is  found  to  be  true  in  all  points  of 
the  revolution. 

The  angular  velocities,  therefore,  which  can  be  measured  very 
accurately  by  the  daily  differences  of  right  ascension  and  declina- 
tion (Art.  132,)  converted  into  corresponding  longitudes,  enable 
us  to  determine  the  different  distances  of  the  earth  from  the  sun 
at  various  points  in  the  orbit. 

166.  Since  the  arcs  described  by  the  earth  in  any  small  times, 
as  in  single  days,  are  inversely  as  the  squares  of  the  distances,  con- 


88 


THE   SUN. 


sequently,  the  distances  are  inversely  as  the  square  roots  of  the  arcs. 
Upon  this  principle,  the  relative  distances  of  the  earth  from  the 
sun,  in  every  point  of  its  revolution,  may  be  easily  calculated. 
Thus,  we  have  seen  that  the  arcs  described  by  the  sun  in  one  day 
at  the  perihelion  and  aphelion  are  as  61  to  57.  Hence  the  distances 
of  the  earth  from  the  sun  at  those  two  points  are  as  -s/57  to  \/61, 
or  as  1  to  1.034.  From  twenty-four  observations  made  with  the 
greatest  care  by  Dr.  Maskelyne  at  the  Royal  Observatory  of 
Greenwich,  the  following  distances  of  the  earth  from  the  sun  are 
determined  for  each  month  in  the  year. 


Time  of  Observation.  Distances. 

January     12-13,  0.98448 

February  17-18,  0.98950 

March        14-15,  0.99622 

April         28-29,  1.00800 

May  15-16,  1.01234 

June  17-18,  1.01654 


Time  of  Observation.  Distances. 

July  18-19,  1.01658 

August  26-27,  1.01042 

September  22-23,  1.00283 

October  24-25,  0.99303 

November  18-20,  0.98746 

December  17-18,  0.98415 


Fig.  33. 


167.  The  angular  velocity  being 
inveipely  as  the  square  of  the  distance 
in  all  parts  of  the  solar  orbit,  it  follows 
that  the  product  of  the  angle  described 
in  any  given  time,  by  the  square  of  the 
distance,  is  always  the  same  constant 
quantity.     For  if  of  two  factors,  A  x 
B,  A  is  increased  as  B  is  diminished, 
the  product  of  A  and  B  is  always  the 
same.     If,  therefore,  from  the  sun  S 
(Fig.  33,)  two  radii  be  drawn  to  T, 

B,  the  extremities  of  the  arc  described  in  one  day,  then  ST2xTB 
gives  the  same  product  in  all  parts  of  the  orbit.* 

168.  The  radius  vector  of  the  solar  orbit  describes  equal  spaces 
in  equal  times,  and  in  unequal  times,  spaces  proportional  to  the  times. 

Let  TB  (Fig.  33,)  be  the  arc  described  by  the  sun  in  one  day ; 
then,  Sector  TSB=£SB  xTB. 


*  TB,  as  seen  from  the  earth,  would  be  projected  into  a  circular  arc,  equal  to  the 
measure  of  the  angle  at  S. 


FIGURE   OF  THE   EARTIl's   ORBIT.  89 

Taking  Sb  as  any  radius,  describe  the  circular  arc  ab,  which  is 
the  measure  of  the  angle  at  S.     Now, 

Sb  :  ab  :  :  SB  :  BT=SBx^  ;  and  substituting  this  value  of  BT 


in  the  above  equation,  we  have  TSB=£SBxSBx      =iSB2x. 

IS0  ho 

But  Sb  is  constant,  and  the  product  of  SB2x«o  is  likewise  constant  ; 
therefore  the  sector  is  always  equal  to  a  constant  quantity,  and 
therefore  the  radius  vector  passes  over  equal  spaces  in  equal 
times.* 

The  sun's  orbit  may  be  accurately  represented  by  taking  some 
point  as  the  perihelion,  drawing  the  radius  vector  to  that  point, 
and,  considering  this  line  as  unity,  drawing  other  radii  making 
angles  with  each  other  such  that  the  included  areas  shall  be  pro- 
portional to  the  times,  and  of  a  length  required  by  the  distance  of 
each  point  as  given  in  the  table  (Art.  166.)  On  connecting  these 
radii,  we  shall  thus  see  at  once  how  little  the  earth's  orbit  departs 
from  a  perfect  circle.  Small  as  the  difference  appears  between 
the  greatest  and  least  distances,  yet  it  amounts  to  nearly  -fa  of  the 
perihelion  distance,  a  quantity  no  less  than  3,000,000  of  miles. 

169.  The  foregoing  method  of  determining  the  figure  of  the 
earth's  orbit  is  founded  on  observation  ;  but  this  figure  is  subject 
to  numerous  irregularities,  the  nature  of  which  cannot  be  clearly 
understood  without  a  knowledge  of  the  leading  principles  of  Uni- 
versal Gravitation.  An  acquaintance  with  these  will  also  be  in- 
dispensable to  our  understanding  the  causes  of  the  numerous  ir- 
regularities, which  (as  will  hereafter  appear)  attend  the  motions 
of  the  moon  and  planets.  To  the  laws  of  universal  gravitation. 
therefore,  let  us  next  apply  our  attention. 

*  Francoeur,  Uran.,  p.  62. 

12 


CHAPTER    III. 

OP  UNIVERSAL    GRAVITATION. 

t 

•  170.  UNIVERSAL  GRAVITATION,  is  that  influence  by  which  every 
body  in  the  universe,  whether  great  or  small,  tends  towards  every 
other,  with  a  force  which  is  directly  as  the  quantity  of  matter,  and 
inversely  as  the  square  of  the  distance. 

As  this  force  acts  as  though  bodies  were  drawn  towards  each 
other  by  a  mutual  attraction,  the  force  is  denominated  attraction  ; 
but  it  must  be  borne  in  mind,  that  this  term  is  figurative,  and  im- 
plies nothing  respecting  the  nature  of  the  force. 

The  existence  of  such  a  force  in  nature  was  distinctly  asserted 
by  several  astronomers  previous  to  the  time  of  Sir  Isaac  Newton, 
but  its  laws  were  first  promulgated  by  this  wonderful  man  in  his 
Principia,  in  the  year  1687.  It  is  related,  that  while  sitting  in  a 
garden,  and  musing  on  the  cause  of  the  falling  of  an  apple,  he 
reasoned  thus  :*  that,  since  bodies  far  removed  from  the  earth  fall 
towards  it,  as  from  the  tops  of  towers,  and  the  highest  mountains, 
why  may  not  the  same  influence  extend  even  to  the  moon ;  and 
if  so,  may  not  this  be  the  reason  why  the  moon  is  made  to  revolve 
around  the  earth,  as  would  be  the  case  with  a  cannon  ball  were 
it  projected  horizontally  near  the  earth  with  a  certain  velocity. 
According  to  the  first  law  of  motion,  the  moon,  if  not  continually 
drawn  or  impelled  towards  the  earth  by  some  force,  would  not 
revolve  around  it,  but  would  proceed  on  in  a  straight  line.  But 
going  around  the  earth  as  she  does,  in  an  orbit  that  is  nearly  cir- 
cular, she  must  be  urged  towards  the  earth  by  some  force,  which, 
in  a  given  time,  may  be  represented  by  the  versed  sine  of  the  arc 
described  in  that  time.  For  let  the  earth  (Fig.  34,)  be  at  E,  and 
let  the  arc  described  by  the  moon  in  one  second  of  time  be  Ab. 
Were  the  moon  influenced  by  no  extraneous  force,  to  turn  her 
aside,  she  would  have  described,  not  the  arc  Ab,  but  the  straight 
line  AB,  and  would  have  been  found  at  the  end  of  the  given  time 

*  Pemberton's  View  of  Newton's  Philosophy. 


UNIVERSAL  GRAVITATION. 


91 


at  B  instead  of  b.  She  therefore  departs  from  the  line  in  which 
she  tends  naturally  to  move,  by  the  line  B6,  which  in  small  angles 
may  be  taken  as  equal  to  the  versed  sine  Aa.  This  deviation 
from  the  tangent  must  be  owing  to 
some  extraneous  force.  Does  this  force 
correspond  to  what  the  force  of  gravi- 
ty exerted  by  the  earth,  would  be  at  the 
distance  of  the  moon?  Now  we  know  the 
distance  of  the  moon  from  the  earth,  and 
of  course  the  circumference  of  her  orbit. 
We  also  know  the  time  of  her  revolu- 
tion around  the  earth.  Hence  we  may 
estimate  the  length  of  the  arc  Ab  de- 
scribed in  one  second ;  and  knowing 
the  arc,  we  can  calculate  its  versed  sine. 
For  the  moon  being  60  times  as  far  from  the  center  of  the  earth, 
as  the  surface  of  the  earth  is  from  the  center,  consequently,  since 
the  force  of  gravity  decreases  as  the  square  of  the  distance  in- 
creases,* the  space  through  which  •  the  moon  would  fall  by  the 

16yV  -  .05  inches. 


force  of  the  earth's  attraction  alone,  would  be 


602 


On  calculating  the  value  of  the  versed  sine  of  the  arc  described  in 
one  second,  it  proves  to  be  the  same.  Hence  gravity,  and  no  other 
force  than  gravity,  causes  the  moon  to  circulate  around  the  earth. 

171.  By  this  process  it  was  discovered  that  the  law  of  gravita- 
tion extends  to  the  moon.  By  subsequent  inquiries  it  was  found 
to  extend  in  like  manner  to  all  the  planets,  and  to  every  member 
of  the  solar  system  ;  and,  finally,  recent  investigations  have  shown 
that  it  extends  to  the  fixed  stars.  The  law  of  gravitation,  there- 
fore, is  now  established  as  the  grand  principle  which  governs  all 
the  motions  of  the  heavenly  bodies.  Hence,  nothing  can  be  more 
deserving  ^  the  attention  of  the  student,  than  the  development  of 
the  results  of  this  universal  law.  A  few  of  them  only  are  all  that 
can  be  exhibited  in  a  work  like  the  present :  their  full  develop- 


*  Natural  Philosophy,  Art.  7.  That  gravity  follows  the  ratio  of  the  inverse  square 
of  the  distance  was,  however,  inferred  by  Newton  from  one  of  Kepler's  Laws,  to  be 
mentioned  hereafter. 


92 


UNIVERSAL  GRAVITATION. 


ment  must  be  sought  for  in  such  great  works  as  the  Mecanique 
Celeste  of  La  Place. 

172.  If  a  body  revolves  about  an  immovable  center  of  force,  and 
is  constantly  attracted  to  it,  it  will  always  move  in  the  same  plane, 
and  describe  areas  about  the  center  proportional  to  the  times.* 

Let  S  (Fig.  35,)  be  the  center  of  force,  and  suppose  a  body  to 
be  projected  at  P  in  the  direction  of  PQR,  and  take  PQ=QR ; 
then,  by  the  first  law  of  motion,  the  body  would  move  uniformly 
in  the  direction  PQR,  and  describe  PQ,  QR,  in  the  same  time,  if 
no  other  force  acted  upon  it.  But  when  the  body  comes  to  Q 

Fig.  35. 


let  a  single  impulse  act  at  S,  sufficient  to  draw  the  body  through 
QV,  in  the  time  it  would  have  described  QR  ;  and  complete  the 
parallelogram  VQRC,  and  the  body  in  the  same  time  will  describe 
QC  ;  therefore,  PQ,  QC,  are  described  in  the  same  time.  But 
the  triangle  SCQ=SRQ=SPQ  ;  that  is,  equal  areas  are  described 
in  equal  times.  For  the  same  reason,  if  a  single  impulse  act  at 
C,  D,  E,  &c.  at  equal  intervals  of  time,  the  several  areas  SPQ, 
SQC,  SCD,  SDE,  &c.  will  all  be  equal  to  each  other.  Now  this 

*  The  learner  will  remark  that  what  has  been  before  proved  (Art.  168,)  respecting 
the  radius  vector  of  the  earth,  is  here  shown  to  hold  good  with  respect  to  every  body 
which  revolves  around  a  center  of  force ;  and  the  same  is  true  of  several  other  propo- 
sitions demonstrated  in  this  chapter. 


UNIVERSAL  GRAVITATION. 


93 


demonstration  is  independent  of  any  particular  dimensions  in  the 
several  triangles,  and  consequently  holds  good  when  they  are 
taken  indefinitely  small,  in  which  case  we  may  consider  the  force 
as  acting,  not  by  separate  impulses,  but  constantly,  causing  the 
body  to  describe  a  curve  around  S.  And  as  no  force  acts  out  of 
the  plane  SPQ,  the  whole  curve  must  lie  in  that  plane ;  that  is, 
the  body  moves  always  in  the  same  plane. 

173.  If  a  body  describes  a  curve  around  a  center  towards  which  it 
tends  by  any  force,  the  angular  velocity  of  the  body  around  that  center 
is  reciprocally  as  the  square  of  the  distance  from  it.* 


Let  ABE  (Fig.  36,)  be  any  curve  de- 
scribed about  the  center  S  ;  draw  SA,  SB, 
to  any  two  points  of  the  curve  A  and  B  ; 
and  let  AD,  BE,  be  described  in  indefi- 
nitely small  equal  times.  Join  SD  and 
SE,  and  with  the  center  S  and  distance 
SD,  describe  a  circle  meeting  SA,  SB,  SE, 
in  F,  G,  H ;  and  with  the  center  S  and 
distance  SE  describe  a  circle  meeting  SB 
inK. 

Because  AD  and  BE  are  described  in 
equal  times,  the  triangles  ASD,  BSE,  are 
equal.     Hence,  (Euc.  15.  6.) 
DF  :  EK  ::  BS  :  ASf  ::  BS2  :  BSxAS  (1) 
SH  :  SE 
(1)DF 
(2)GH 
.-.  DF 


Fig.  36. 


GH:EK 

Hence, 


:SF 
BS2 

:AS2 
:GH 


SA2:BSxAS  (2) 


SE  ::SA:SB 

:EK:BSxAS 

:EK:BSxAS 

:  BS2  :  AS2. 

But  DF  and  GH  measure  the  respective  angular  velocities  at 
A  and  B,  while  AS  and  BS  represent  the  distance  at  the  same 
points.  Therefore  the  angular  velocities  are  reciprocally  as  the 
squares  of  the  distances.  J 

174.  In  the  same  curve,  the  velocity,  at  any  point  of  the  curve, 

*  It  will  be  remarked  that  this  is  a  general  proposition,  of  which  article  165  affords 
A  particular  example. 

t  DF  and  EK  are  considered  as  the  altitudes  of  the  triangles  respectively, 
f  Stewart's  Phys.  and  Math.  Essays. 


94  UNIVERSAL  GRAVITATION. 

varies  inversely  as  the  perpendicular  drawn  from  the  center  of 
force  to  the  tangent  at  that  point. 

Draw  SY  (Fig.  35,)  perpendicular  to  QP  produced  ;  then  the 
area   SPQ=iPQ  x  SY,   which   varies   as   PQ  x  SY  /.  PQ  a 


in  the  curve  described  from  P,  with  a  constant  force,  SY  becomes 
a  perpendicular  to  the  tangent  to  the  curve.  But  by  article 
172,  the  area  described  in  a  given  time  is  constant.  Therefore 

SPQ  is  constant,  and  V  a  -  ;  that  is,  the  velocity  varies  inverse- 

SY 

ly  as  the  perpendicular  upon  the  tangent.  Hence,  the  velocity  of 
a  revolving  body  increases  as  it  approaches  the  center  of  force. 

175.  If  equal  areas  be  described  about  a  center  in  equal  times. 
the  force  must  tend  towards  that  center. 

Let  SPQ  (Fig.  35,)=SQC  ;  now  SPQ-SQR  /.  SQC-SQR.-. 
CR  is  parallel  to  QS.  Complete  the  parallelogram  QRCV,  and 
by  the  supposition  the  body  describes  QC,  in  consequence  of  the 
impulse  at  Q,  and  it  would  have  described  QR  if  no  such  impulse 
had  acted  ;  therefore  QV  must  represent  that  motion  impressed 
at  Q,  which,  in  conjunction  with  the  motion  QR,  can  make  a  body 
describe  QC,  and  QV  is  directed  to  S. 

176.  Now  it  appears  from  article  168,  that  it  is  a  fact,  derived 
from  observation,  that  the  earth's  radius  vector  describes  equal 
areas  in  equal  times  ;  and  by  similar  observations  the  same  is 
found  to  be  true  of  each  of  the  primary  planets  about  the  sun, 
and  of  each  of  the  satellites  about  its  primary.     Hence,  it  is  in- 
ferred, that  the  primary  planets  all  gravitate  towards  the  sun,  and 
that  the  secondary  planets  all  gravitate  towards  their  respective 
primaries. 

It  has  further  been  established  by  observation,  (Art.  162,)  that 
the  planetary  orbits  are  ellipses  ;  and  hence  the  application  of  the 
principles  of  gravitation,  so  far  as  respects  the  sun  and  planets, 
may  be  confined  to  the  consideration  of  the  motion  of  a  body  in 
an  elliptical  orbit. 

177.  The  distance  of  any  planet  from  the  sun  at  any  point  in  its 


UNIVERSAL  GRAVITATION. 


95 


orbit,  is  to  its  distance  from  the  superior  focus,  as  the  square  of  its 
velocity  at  its  mean  distance  from  the  sun,  is  to  the  square  of  its  ve- 
locity at  the  given  point. 

Let  ADBE  (Fig.  37,)  be  the  orbit  of  a  planet,  S  the  focus  in 
which  the  sun  is  placed,  AB  the  transverse  and  DE  the  conjugate 
axis,  C  the  center,  and  F  the  superior  focus.  Let  the  planet  be 
any  where  at  P ;  and  draw  a  tangent  to  the  orbit  at  P,  on  which 
from  the  foci  let  fall  the  perpendiculars  SG,  FH.  Draw  also  DK 
touching  the  orbit  in  D,  and  let  SK  be  perpendicular  to  it.  Let 

Fig.  37. 

N 


the  velocity  of  the  planet  when  at  the  mean  distance  at  D— C,  and 
when  at  P=V.  Join  SP,  FP.  Then  (Art.  174,)  the  velocity  at 
D  is  to  the  velocity  at  P,  as  SG  to  SK ;  that  is, 

C:V::SG:DC. 
C2  :  V2  : :  SG2  :  DC2. 

But  because  the  triangles  SGP,  FHP,  are  equiangular,  having 
right  angles  at  G  and  H,  and  also,  from  the  nature  of  the  ellipse, 
the  angles  SPG,  FPH,  equal, 

SP  :  PF  : :  SG  :  FH  :  :  SG2  :  CD2=FHxSG 
.-.  SP:PF::C2:V2 

178.  If  of  two  bodies  gravitating  to  the  same  center,  one  descends 
in  a  straight  line,  and  the  other  revolves  in  a  curve  ;  then,  if  the  ve- 
locities of  these  bodies  are  equal  in  any  one  case,  when  they  are 


96 


UNIVERSAL  GRAVITATON. 


Fig.  38. 


equally  distant  from  the  center,  they  will  always  be  equal  when  they 
are  equally  distant  from  it. 

Let  ABC  (Fig.  38,)  be  a  curve  which  a  body 
describes  about  a  center  S  to  which  it  gravi- 
tates, while  another  body  descends  in  a 
straight  line  AS  to  that  center.  Let  BC  be 
any  arc  of  the  curve  ABC,  and  let  BD,  CH, 
be  arcs  of  circles  described  from  the  center 
S,  intersecting  the  line  AS  in  D  and  H. 
From  the  center  S  describe  the  arc  bd,  in- 
definitely near  to  BD,  and  draw  E/  perpen- 
dicular to  ~Bb.  Then,  because  the  distances 
SD  and  SB  are  equal,  the  forces  of  gravity 
at  D  and  B  are  also  equal.  Let  these  forces 
be  expressed  by  the  equal  lines  Dd  and  BE  ; 
and  let  the  force  BE  be  resolved  into  the 
forces  E/"  and  B/*.  The  force  E/,  acting  at 
right  angles  to  the  path  of  the  body,  will  not  affect  its  velocity  in 
that  path,  but  will  only  draw  it  aside  from  a  rectilinear  course  and 
make  it  proceed  in  the  curve  B&C.  But  the  other  force  B/",  acting 
in  the  direction  of  the  course  of  the  body,  will  be  wholly  employed 
in  accelerating  it.  And  because  B  and  b  are  indefinitely  near  to 
each  other,  and  likewise  D  and  d,  the  accelerating  force  from  B  to 
b  and  from  D  to  d,  may  be  considered  as  acting  uniformly. 
Therefore,  the  accelerations  of  the  bodies  in  D  and  B,  produced 
in  equal  times,  are  as  the  lines  Dd,  B/";  and  hence,  putting  d  for 
the  increment  of  velocity  at  d,  and/  for  the  increment  of  velocity 
at/, 

d:f:i  Dd  or  BE  :B/.  (1) 

And  because  the  angle  at  E  is  a  right  angle, 


Hence,  BE :  B/: :  v/B6  :  x/B/  (2) 

And,  (1)  and  (2),  d :/: :  x/B6  :  x/B/  (3) 

But,  putting  6  for  the  velocity  at  &,  and  observing  that,  in  falling 
bodies,  the  velocities  are  as  the  square  roots  of  the  spaces, 
bifn  x/B6:  VB/"-  (4) 

Therefore,  (3)  and  (4),  b:f::d:f.:  b=d  ;  that  is,  the  velocity  at 
b  equals  the  velocity  at  d.    And,  since  the  same  reasoning  holds 


UNIVERSAL    GRAVITATION.  97 

for  successive  points  that  may  be  taken  at  equal  distances  from  B 
and  D,  therefore,  if  of  two  bodies,  &c.* 

179.  The  law  according  to  which  the  planets  gravitate  is  such,  that 
any  body  under  the  influence  of  the  same  force,  and  falling  direct  to 
the  sun,  will  have  its  velocity  at  any  point  equal  to  a  constant  velocity 
multiplied  into  the-  square  root  of  the  distance  it  has  fallen  through, 
divided  by  the  square  root  of  the  distance  between  the  body  and  the 
mrfs  center. 

Suppose  a  planet  to  revolve  in  the  elliptical  orbit  APB  (Fig.  37); 

at  A,  the  higher  apsis,  the  velocity  V=C       ~P  (Art.  177)  ;  or 


if  AN,  in  the  axis  produced=AF,  v=c  Let  a  body  at 

A  begin  to  descend  towards  S  with  this  velocity,  then  if  SL=SP, 
the  velocity  of  the  planet  at  P  will  be  the  same  as  that  of  the  fall- 
ing body  at  L,  (Art.  178.)  But  the  velocity  of  the  planet  at  P  is 

\PS/  ~^  (sT~)  ^Ut  ^  verity  *s  equal  to  the  constant  ve- 
locity expressed  by  C,  multiplied  into  the  square  root  of  NL,  the 
distance  fallen  through,J  divided  by  the  square  root  of  LS,  the 
distance  between  the  body  and  the  sun's  center.§ 

180.  The  force  with  which  any  planet  gravitates  to  the  sun,  is  in- 
versely as  the  square  of  its  distance  from  the  sun's  center. 

Let  C  (Fig.  39,)  be  the  center  to  which  the  falling  body  gravi- 
tates, A  the  point  from  which  it  begins  to  fall,  and  its  velocity  at 
any  point  B,  is  to  its  velocity  in  the  point  G,  which  bisects  AC,  as 

(BC/  :  L"  Let  DEF  be  a  curve  such  that  if  AD  be  an  ordinate 

or  a  perpendicular  to  AC,  meeting  the  curve  in  D,  and  BE  any  other 

*  Principia,  Lib.  i,  Pr.  40.     Stewart's  Math,  and  Phys.  Essays,  Pr.  13. 
t  For  SN=AB=SP-j-PF=:SP-[-NL  .-.  PF=NL. 

*  That  NL  (=PF)  is  the  distance  fallen  through  to  acquire  the  velocity  at  P,  is  de- 
monstrated by  writers  on  Central  Forces.   (See  Vince,  Syst.  Ast.,  Art.  823.) 

§  Playfair,  Phys.  Ast. 

||  For,  denoting  the  velocity  at  B  by  V,  and  the  velocity  at  G  by  V, 


13 


98  UNIVERSAL    GRAVITATION. 

ordinate,  AD  is  to  BE  as  the  force  at  A  to  the  force  at  B,  then 
will  twice  the  area  ABED  be  equal  to  the  Fig.  39 


square  of  the  velocity  which  the  body  has 
acquired  in  B.*     If  therefore  the   velocity      33 
at  B  be  V,  that  at  the  middle  point  G  being  c,       l 

=c  (£?)*,  and  therefore  2  ABED-c2.  £?  ; 


and   since  AB  =AC  -  BC,  2  ABED  =  c>. 

AC-BC     9/AC      \    „ 

—  —  -  —  —&  I  -—  —  1  1.  For  the  same  reason, 
BO  \BC       / 

if  be  be  drawn  indefinitely  near  to  BE,  2AbeD 
*  \  f~*       \ 

=c2  (  —  —  —  1  ),  and  therefore  the  difference  of 
V  0C        / 

these  areas,  or  2RbeE,  that  is, 


D 


AC\     0AC(BC-6C)     2ACxB6  ,xr, 
-BcH      BCX6C     =C—*-  WhereforMmdmgby 


;  now  c2  and  AG  are  constant 

quantities,  therefore  EB  varies  inversely  as  BC2.  But  EB  repre- 
sents the  force  of  gravity  at  B,  and  BC  the  distance  from  the 
sun.  Therefore,  the  force  of  gravity  of  a  planet  in  different  parts 
of  its  orbit,  is  inversely  as  the  square  of  its  distance  from  the  sun. 

181.  The  line  CG  is  the  same  with  the  mean  distance  of  the 
planet  in  an  orbit  of  which  AC  is  the  length  of  the  transverse  axis  ; 
and  if  the  gravitation  at  that  distance  =F,  and  the  mean  distance 


itself=c,  then  since  EB-c2-,  F=c2x-=-,  or  «F=c2. 

«*    a 


*  This  principle  is  demonstrated  by  the  aid  of  Fluxions  as  follows : 

By  construction,  BE  is  proportional  to  the  force  at  B=^-,   v  being   the  velocity 

which  the  moving  body  has  acquired  at  B,  and  t  the  time  of  the  descent  from  A  to  B. 
Now  B6  is  the  momentary  increment  of  BA  the  space,  and  therefore=ud* ;  therefore 
BExB6=wd0.  And  2BExB&r=2t)d».  But  BExB6  is  the  momentary  increment  of 
the  area  ABED,  and  %vdv  is  the  momentary  increment  of  t)2 ;  therefore  the  square  of 
the  velocity  of  the  moving  body,  and  twice  the  area  of  ABED,  increase  at  the  same 
rate,  and  begin  to  exist  at  the  same  time  ;  therefore  they  are  equal.  (See  Playfair'a 
Outlines,  Mechanics,  Art.  96.) 

t  iC  being  ultimately  equal  to  BC. 


UNIVERSAL   GRAVITATION.  99 

182.  The  squares  of  the  times  of  revolution  of  any  two  planets, 
are  as  the  cubes  of  their  mean  distances  from  the  sun. 

If  a  be  the  mean  distance,  or  the  semi-transverse  axis,  b  the 
semi-conjugate,  then  #a&=area  of  the  orbit.*  But  as  c  is  the  ve- 
locity at  the  mean  distance,  or  the  elliptic  arch  which  the  planet 
moves  over  in  a  second  when  it  is  at  D,  (Fig.  37.)  the  vertex  of  the 
conjugate  axis,  therefore  ^bc  is  the  area  described  in  that  second 
by  the  radius  vector  ;  and  since  the  area  is  the  same  for  every 
second  of  the  planet's  revolution  (Art.  172,)  therefore  the  area  of 
the  orbit  divided  by  %bc  will  give  the  number  of  seconds  in 


which  the  revolution  is  completed,  which=     -  =  -  ;    or,  since 

\bc       c 

c*  =  aF,  (Art.  181,)  the  time  of  a  revolution  =  —  ==%<K\/  =. 

vaF 

Hence,  let  t,  t',  be  the  times  of  revolutions  for  two  different  plan- 
ets, of  which  the  mean  distances  are  a,  a',  and  the  force  of  gravity 

at  those  distances  F,  P.   Then  titfiiZ*  \/£-  -  2*\/Y>:  : 


?s-,  or  $  :  t'z  :  :  a3  :  a'3.  That  is,  the  squares  of  the  times  are  as  the 
or 

cubes  of  the  mean  distances  ;  or,  since  the  major  axes  of  the  or- 
bits are  double  the  mean  distances,  the  squares  of  the  times  are  as 
the  cubes  of  the  major  axes. 

183.  This  is  one  of  Kepler  's  three  great  Laws,  which,  taken  in 
connexion,  are  as  follows  : 

1.  The  orbits  of  all  the  planets  are  ellipses,  the  sun  occupying  the 
common  focus.     (Art.  176.) 

2.  The  radius  vector  of  any  planet  describes  areas  proportional 
to  the  times.     (Art.  172.) 

3.  The  squares  of  the  periodical  times  are  as  the  cubes  of  the  ma- 
jor axes  of  the  orbits.     (Art.  182.) 

These  great  and  fundamental  principles  of  the  planetary  mo- 
tions, were  discovered  by  the  illustrious  Kepler  by  long  and  as- 
siduous study  of  the  observations  made  by  Tycho  Brahe,  and 

*  Day's  Mensuration. 


100  UNIVERSAL   GRAVITATION. 

hence  he  has  been  called  the  legislator  of  the  skies.     They,  there 
fore,  became  known  as  facts,  before  they  were   demonstrated 
mathematically.     The  glory  of  this  achievement  was   reserved 
for  Newton,  who  proved  that  they  were  necessary  results  of  the 
law  of  universal  gravitation. 

MOTION    IN    AN    ELLIPTICAL    ORBIT. 

184.  Having  now  acquired  some  knowledge  of  the  law  of  uni- 
versal gravitation,  let  us  next  endeavor  to  gain  a  just  conception 
of  the  forces  by  which  the  planets  are  made  to  revolve  in  their 
orbits  about  the  sun.  In  obedience  to  the  first  law  of  motion, 
every  moving  body  tends  to  move  in  a  straight  line  ;  and  were  not 
the  planets  deflected  continually  towards  the  sun  by  the  force  of 
attraction,  these  bodies  as  well  as  others  would  move  forward  in 
a  rectilineal  direction.  We  call  the  force  by  which  they  tend  to 
such  a  direction  the  projectile  force,  because  its  effects  are  the 
same  as  though  the  body  were  originally  projected  from  a  certain 
point  in  a  certain  direction.  It  is  an  interesting  problem  for  me- 
chanics to  solve,  what  was  the  nature  of  the  impulse  originally 
given  to  the  earth,  in  order  to  impress  upon  it  its  two  motions,  the 
one  around  its  own  axis,  the  other  around  the  sun  ?  If  struck  in 
the  direction  of  its  center  of  gravity  it  might  receive  a  forward 
motion,  but  no  rotation  on  its  axis.  It  must,  therefore,  have  been 
impelled  by  a  force,  whose  direction  did  not  pass  through  its  cen- 
ter of  gravity.  Bernouilli,  a  celebrated  mathematician,  has  calcu- 
lated that  the  impulse  must  have  been  given  very  nearly  in  the 
direction  of  the  center,  the  point  of  projection  being  only  the  165th 
part  of  the  earth's  radius  from  the  center.*  This  impulse  alone 
would  cause  the  earth  to  move  in  a  right  line :  gravitation  towards 
the  sun  causes  it  to  describe  an  orbit.  Thus  a  top  spinning  on  a 
smooth  plane,  as  that  of  glass  or  ice,  if  impelled  in  a  direc- 
tion not  passing  through  the  center  of  gravity,  may  be  made  to 
imitate  the  two  motions  of  the  earth,  especially  if  the  experiment 
is  tried  in  a  concave  surface  like  that  of  a  large  bowl.  The  re- 
sistance occasioned  by  the  surface  on  which  the  top  moves,  and 

*  Francceur,  Uran.  p.  49 . 


UNIVERSAL    GRAVITATION. 


101 


that  of  the  air,  will  generally  destroy  the  force  of  projection  and 
cause  the  top  to  revolve  in  a  smaller  and  smaller  orbit ;  but  the 
earth  meets  with  no  such  resistance,  and  therefore  makes  both  her 
days  and  years  of  the  same  length  from  age  to  age.  A  body, 
therefore,  revolving  in  an  orbit  about  a  center  of  attraction,  is 
constantly  under  the  influence  of  two  forces, — the  projectile  force, 
which  tends  to  carry  it  forward  in  a  straight  line  which  is  a  tan- 
gent to  its  orbit,  and  the  centripetal  force,  by  which  it  tends  to- 
wards the  center. 

185.  The  most  simple  example  we  have  of  the  combined  action 
of  these  two  forces  is  the  motion  of  a  missile  thrown  from  the 
hand,  or  of  a  ball  fired  from  a  cannon.  It  is  well  known  that  the 
particular  form  of  the  curve  described  by  the  projectile,  in  either 
case,  will  depend  upon  the  velocity  with  which  it  is  thrown.  In 
each  case  the  body  will  begin  to  move  in  the  line  of  direction  in 
which  it  is  projected,  but  it  will  soon  be  deflected  from  that  line 
towards  the  earth.  It  will  however  continue  nearer  to  the  line  of 
projection  as  the  velocity  of  projection  is  greater.  Thus  let  AB 

Fig.  40. 


B 


(Fig.  40,)  perpendicular  to  AC  represent  the  line  of  projection. 
The  body  will,  in  every  case,  commence  its  motion  in  the  line  AB, 
which  will  therefore  be  the  tangent  to  the  curve  it  describes  ;  but 
if  it  be  thrown  with  a  small  velocity,  it  will  soon  depart  from  the 
tangent,  describing  the  line  AD ;  with  a  greater  velocity  it  will 
describe  a  curve  nearer  to  the  tangent,  as  AE ;  and  with  a  still 
greater  velocity  it  will  describe  the  curve  AF. 

As  an  example  of  a  body  revolving  in  an  orbit  under  the  influ- 
ence of  two  forces,  suppose  a  body  placed  at  any  point  P  (Fig.  40') 
above  the  surface  of  the  earth,  and  let  PA  be  the  direction  of  the 
earth's  center ;  that  is,  a  line  perpendicular  to  the  horizon.  If  the 


102 


UNIVERSAL   GRAVITATION. 


body  were  allowed  to  move  without  receiving  any  impulse,  it 
would  descend  to  the  earth  in  the  direction  PA  with  an  accelerated 
motion.  But  suppose  that,  at  the  moment  of  its  departure  from 
P,  it  receives  a  blow  in  the  direction  PB,  which  would  carry  it  to 
B  in  the  time  the  body  would  fall  from  P  to  A  ;  then,  under  the  in- 
fluence of  both  forces,  it  would  descend  along  the  curve  PD.  If 
a  stronger  blow  were  given  to  it  in  the  direction  PB,  it  would  de- 
scribe a  larger  curve,  PE ;  or,  finally,  if  the  impulse  were  suffi- 
ciently strong,  it  would  circulate  quite  around  the  earth,  and  re- 
turn again  to  P,  describing  the  circle  PFG.  With  a  velocity  of 
projection  still  greater,  it  would  describe  an  ellipse,  PIK ;  and  if 
the  velocity  were  increased  to  a  certain  degree,  the  figure  would 
become  a  parabola  or  hyperbola  LMP,  and  never  return  into 
itself. 

186.  In  figure  41,  suppose  the  planet  to  have  passed  the  point  C 
with  so  small  a  velocity,  that  the  attraction  of  the  sun  bends  its 
path  very  much,  and  causes  it  immediately  to  begin  to  approach 
towards  the  sun ;  the  sun's  attraction  will  increase  its  velocity  as 
it  moves  through  D,  E,  and  F.  For  the  sun's  attractive  force  on 
the  planet,  when  at  D,  is  acting  in  the  direction  DS,  and,  on  account 
of  the  small  inclination  of  DE  to  DS,  the  force  acting  in  the  line 
DS  helps  the  planet  forward  in  the  path  DE,  and  thus  increases 
its  velocity.  In  like  manner  the  velocity  of  the  planet  will  be  con- 
tinually increasing  as  it  passes  through  D,  E,  and  F ;  and  though 
the  attractive  force,  on  account  of  the  planet's  nearness,  is  so  muca 
increased,  and  tends,  therefore,  to  make  the  orbit  more  curved, 


UNIVERSAL   GRAVITATION. 


103 


yet  the  velocity  is  also  so  much  increased,  that  the  orbit  is  not 
more  curved  than  before.  The  same  increase  of  velocity  occa- 
sioned by  the  planet's  approach  to  the  sun,  produces  a  greater  in- 
crease of  centrifugal  force  which  carries  it  off  again.  We  may 
see  also  why,  when  the  planet  has  Fig.  41. 

reached  the  most  distant  parts  of  its 
orbit,  it  does  not  entirely  fly  off,  and 
never  return  to  the  sun.  For  when 
the  planet  passes  along  H,  K,  A,  the 
sun's  attraction  retards  the  planet, 
just  as  gravity  retards  a  ball  rolled  up 
hill ;  and  when  it  has  reached  C,  its 
velocity  is  very  small,  and  the  attrac- 
tion at  the  center  of  force  causes  a 
a  great  deflection  from  the  tangent, 
sufficient  to  give  its  orbit  a  great  cur- 
vature, and  the  planet  turns  about,  returns  to  the  sun,  and  goes 
over  the  same  orbit  again.*  As  the  planet  recedes  from  the  sun, 
its  centrifugal  force  diminishes  faster  than  the  force  of  gravity,  so 
that  the  latter  finally  preponderates^ 

187.  We  may  imitate  the  motion  of  a  body  in  its  orbit  by  sus- 
pending a  small  ball  from  the  ceiling  by  a  long  string.  The  ball 
being  drawn  out  of  its  place  of  rest,  (which  is  directly  under  the 
point  of  suspension,)  it  will  tend  constantly  towards  the  ^ame 
place  by  a  force  which  corresponds  to  the  force  of  attraction  of  a 
central  body.  If  an  assistant  stands  under  the  point  of  suspen- 
sion, his  head  occupying  the  place  of  the  ball  when  at  rest,  the 
ball  may  be  made  to  revolve  about  his  head  as  the  earth  or  any 
planet  revolves  about  the  sun.  By  projecting  the  ball  in  different 
directions,  and  with  different  degrees  of  velocity,  we  may  make 
it  describe  different  orbits,  exemplifying  principles  which  have 
been  explained  in  the  foregoing  propositions. 

»  Airy. 

t  The  centrifugal  force  varies  inversely  as  the  cube  of  the  distance,  while  the  forco 
of  gravity  is  inversely  as  the  square.  The  centrifugal  force,  therefore,  increases  faster 
than  the  force  of  gravity  as  a  body  is  approaching  the  sun,  and  decreases  faster  as  the 
body  recedes  from  the  sun.  (See  M.  Stewart's  Phys.  and  Math.  Tracts,  Prop.  8.") 


CHAPTER  IV. 

PRECESSION    OF  THE  EQUINOXES NUTATION ABERRATION MOTION 

OP  THE  APSIDES MEAN  AND  TRUE  PLACES  OF  THE  SUN. 

188.  THE  PRECESSION  OF  THE  EQUINOXES,  is  a  slow  but  continual 
shifting  of  the  equinoctial  points  from  east  to  west. 

Suppose  that  we  mark  the  exact  place  in  the  heavens,  where, 
during  the  present  year,  the  sun  crosses  the  equator,  and  that  this 
point  is  close  to  a  certain  star ;  next  year  the  sun  will  cross  the 
equator  a  little  way  westward  of  that  star,  and  so  every  year  a 
little  further  westward,  until,  in  a  long  course  of  ages,  the  place 
of  the  equinox  will  occupy  successively  every  part  of  the  ecliptic, 
until  we  come  round  to  the  same  star  again.  As,  therefore,  the 
sun,  revolving  from  west  to  east  in  his  apparent  orbit,  comes 
round  towards  the  point  where  it  left  the  equinox,  it  meets  the 
equinox  before  it  reaches  that  point.  The  appearance  is  as  though 
the  equinox  goes  forward  to  meet  the  sun,  and  hence  the  phenom- 
enon is  called  the  Precession  of  the  Equinoxes,  and  the  fact  is 
expressed  by  saying  that  the  equinoxes  retrograde  on  the  ecliptic, 
until  the  line  of  the  equinoxes  makes  a  complete  revolution  from 
east  to  west.  The  equator  is  conceived  as  sliding  westward  on 
the  ecliptic,  always  preserving  the  same  inclination  to  it,  as  a  ring 
placed  at  a  small  angle  with  another  of  nearly  the  same  size, 
which  remains  fixed,  may  be  slid  quite  around  it,  giving  a  cor- 
responding motion  to  the  two  points  of  intersection.  It  must  be 
observed,  however,  that  this  mode  of  conceiving  of  the  precession 
of  the  equinoxes  is  purely  imaginary,  and-  is  employed  merely  for 
the  convenience  of  representation. 

189.  The   amount  of  precession  annually  is  50."  1 ;   whence, 
since  there  are  3600"  in  a  degree,  and  360°  in  the  whole  circum- 
ference, and  consequently,  1296000",  this  sum  divided  by  50.1 
gives  25868  years  for  the  period  of  a  complete  revolution  of  the 
equinoxes. 


PRECESSION    OF    THE    EQUINOXES.  105 

190.  Suppose  now  we  fix  to  the  center  of  each  of  the  two 
rings  (Art.  188)  a  wire  representing  its  axis,  one  corresponding  to 
the  axis  of  the  ecliptic,  the  other  to  that  of  the  equator,  the  ex- 
tremity of  each  being  the  pole  of  its  circle.  As  the  ring  deno- 
ting the  equator  turns  round  on  the  ecliptic,  which  with  its  axis 
remains  fixed,  it  is  easy  to  conceive  that  the  axis  of  the  equator 
revolves  around  that  of  the  ecliptic,  and  the  pole  of  the  equator 
around  the  pole  of  the  ecliptic,  and  constantly  at  a  distance  equal 
to  the  inclination  of  the  two  circles.  To  transfer  our  conceptions 
to  the  celestial  sphere,  we  may  easily  see  that  the  axis  of  the  diur- 
nal sphere,  (that  of  the  earth  produced,  Art.  28,)  would  not  have 
its  pole  constantly  in  the  same  place  among  the  stars,  but  that  this 
pole  would  perform  a  slow  revolution  around  the  pole  of  the 
ecliptic  from  east  to  west,  completing  the  circuit  in  about  26,000 
years.  Hence  the  star  which  we  now  call  the  pole  star,  has  not 
always  enjoyed  that  distinction,  nor  will  it  always  enjoy  it  here- 
after. When  the  earliest  catalogues  of  the  stars  were  made,  this 
star  was  12°  from  the  pole.  It  is  now  1°  24',  and  will  approach 
still  nearer ;  or,  to  speak  more  accurately,  the  pole  will  come  still 
nearer  to  this  star,  after  which  it  will  leave  it,  and  successively 
pass  by  others.  In  about  13,000  years,  the  bright  star  Lyra, 
which  lies  on  the  circle  of  revolution  opposite  to  the  present  pole 
star,  will  be  within  5°  of  the  pole,  and  will  constitute  the  Pole 
Star.  As  Lyra  now  passes  near  our  zenith,  the  learner  might 
suppose  that  the  change  of  position  of  the  pole  among  the  stars, 
would  be  attended  with  a  change  of  altitude  of  the  north  pole 
above  the  horizon.  This  mistaken  idea  is  one  of  the  many  mis- 
apprehensions which  result  from  the  habit  of  considering  the 
horizon  as  a  fixed  circle  in  space.  However  the  pole  might  shift 
its  position  in  space,  we  should  still  be  at  the  same  distance  from 
it,  and  our  horizon  would  always  reach  the  same  distance  be- 
yond it. 

191.  The  precession  of  the  equinoxes  is  an  effect  of  the  spheroidal 
figure  of  the  earth,  and  arises  from  the  attraction  of  the  sun  and 
moon  upon  the  excess  of  matter  about  the  earths  equator. 

Were  the  earth  a  perfect  sphere  the  attractions  of  the  sun  and 
moon  upon  the  earth  would  be  in  equilibrium  among  themselves, 

14 


106 


THE    SUN. 


But  if  a  globe  were  cut  out  of  the  earth,  (taking  half  the  polar 
diameter  for  radius,)  it  would  leave  a  protuberant  mass  of  matter 
in  the  equatorial  regions,  which  may  be  considered  as  all  collected 
into  a  ring  resting  on  the  earth.  The  sun  being  in  the  ecliptic, 
while  the  plane  of  this  ring  is  inclined  to  the  ecliptic  23°  28',  of 
course  the  action  of  the  sun  is  oblique  to  the  ring,  and  may  be 
resolved  into  two  forces,  one  in  the  plane  of  the  equator,  and  the 
other  perpendicular  to  it.  The  latter  only  can  act  as  a  disturbing 
force,  and  tending  as  it  does  to  draw  down  the  ring  to  the  ecliptic, 
the  ring  would  turn  upon  the  line  of  the  equinoxes  as  upon  a 
hinge,  and  dragging  the  earth  along  with  it,  the  equator  would 
ultimately  coincide  with  the  ecliptic  were  it  not  for  the  revolution 
of  the  earth  upon  its  axis.  This  may  be  better  understood  by  the 
aid  of  a  diagram.  Let  TAB  (Fig.  42,)  represent  the  equator, 

Fig.  42. 
E 


TED  the  ecliptic,  and  AD  the  solstitial  colure.  Let  AB  be  the 
movement  of  rotation  for  a  very  short  time,  being  of  course  in  the 
order  of  the  signs  and  in  the  direction  of  the  equator.  Let  BC  be 
the  movement  produced  by  the  disturbing  force  of  the  sun  in  the 
same  time.  The  point  A  will  describe  the  diagonal  AC,  the  equa- 
tor will  take  the  inclined  situation  CAT' ;  the  equinoctial  point 
will  retrograde  from  T  to  T' ;  the  colure  AD  will  take  the  posi- 
tion AE,  while  the  inclination  of  the  two  planes,  that  is,  the  ob- 
liquity of  the  ecliptic,  will  remain  nearly  the  same.* 

192.  The  moon  conspires  with  the  sun  in  producing  the  pre- 
cession of  the  equinoxes,  its  effect,  on  account  of  its  nearness  to 
the  earth,  being  more  than  double  that  of  the  sun,  or  as  7  to  3. 
The  planets  likewise,  by  their  attraction,  produce  a  small  effec 

»  Delambre,  t.  3,  p.  145.    Playfair's  Outlines,  2,  308. 


PRECESSION    OF    THE    EQUINOXES.  107 

upon  the  equatorial  ring,  but  the  result  is  slightly  to  diminish  the 
amount  of  precession.  The  whole  effect  of  the  sun  and  moon 
being  50."41,  that  of  the  planets  is  0.31,  leaving  the  actual  amount 
of  precession  50."!.* 

This  effect  is  not  to  be  imagined  as  taking  place  merely  at  the 
time  of  the  equinoxes,  but  as  resulting  constantly  from  the  action 
of  the  sun  and  moon  on  the  equatorial  ring,  and  at  every  revolu- 
tion of  this  ring  along  with  the  earth  on  its  axis.  Conceive  of 
any  point  in  the  ring,  and  follow  it  round  in  the  diurnal  revolution, 
and  it  will  be  seen  that  that  point,  in  consequence  of  the  attrac- 
tion of  the  sun  and  moon,  will  be  made  to  cross  the  ecliptic  a  little 
further  westward  than  on  the  preceding  day. 

193.  The  time  occupied  by  the  sun  in  passing  from  the  equinoc- 
tial point  round  to  the  same  point  again,  is  called  the  TROPICAL  YEAR. 
As  the  sun  does  not  perform  a  complete  revolution  in  this  inter- 
val, but  falls  short  of  it  50."  1,  the  tropical  year  is  shorter  than  the 
sidereal  by  20m.  20s.  in  mean  solar  time,  this  being  the  time  of 
describing  an  arc  of  50."  1  in  the  annual  revolution.!  The 
changes  produced  by  the  precession  of  the  equinoxes  in  the  ap- 
parent places  of  the  circumpolar  stars,  have  led  to  some  interest- 
ing results  in  chronology.  In  consequence  of  the  retrograde  mo- 
tion of  the  equinoctial  points,  the  signs  of  the  ecliptic  (Art.  35,) 
do  not  correspond  at  present  to  the  constellations  which  bear  the 
same  names,  but  lie  about  one  whole  sign  or  30°  westward  of 
them.  Thus,  that  division  of  the  ecliptic  which  is  called  the  sign 
Taurus,  lies  in  the  constellation  Aries,  and  the  sign  Gemini  in  the 
constellation  Taurus.  Undoubtedly,  however,  when  the  ecliptic 
was  thus  first  divided,  and  the  divisions  named,  the  several  con- 
stellations lay  in  the  respective  divisions  which  bear  their  names. 
How  long  is  it,  then,  since  our  zodiac  was  formed  ? 

50."1  :  1  year  ::  30°(=108000")  :  2155.6  years. 

The  result  indicates  that  the  present  divisions  of  the  zodiac 
were  made  soon  after  the  establishment  of  the  Alexandrian  school 
of  astronomy.  (Art  2.) 

*  Francoeur,  Uran.  162.  t  59'  8."3  :  24h.  : :  50."1  :  20m.  20s. 


108  THE   SUN. 


NUTATION. 

194.  NUTATION  is  a  vibratory  motion  of  the  earth's  axis,  arising 
from  periodical  fluctuations  in  the  obliquity  of  the  ecliptic. 

If  the  sun  and  moon  moved  in  the  plane  of  the  equator,  there 
would  be  no  precession,  and  the  effect  of  their  action  in  producing 
it  varies  with  their  distance  from  that  plane.  Twice  a  year,  there- 
fore, namely,  at  the  equinoxes,  the  effect  of  the  sun  is  nothing ; 
while  at  the  solstices  the  effect  of  the  sun  is  a  maximum.  On 
this  account,  the  obliquity  of  the  ecliptic  is  subject  to  a  semi-an- 
nual variation,  since  the  sun's  force  which  tends  to  produce  a 
change  in  the  obliquity  is  variable,  while  the  diurnal  motion  of 
the  earth  which  prevents  the  change  from  taking  place,  is  con- 
stant. Hence  the  plane  of  the  equator  is  subject  to  an  irregular 
motion  which  is  called  the  Solar  Nutation.  The  name  is  derived 
from  the  oscillatory  motion  communicated  by  it  to  the  earth's  axis, 
while  the  pole  of  the  equator  is  performing  its  revolution  around 
the  pole  of  the  ecliptic  (Art.  190.)  The  effect  of  the  sun  however 
is  less  than  that  of  the  moon,  in  the  ratio  of  2  to  5.  By  the  nuta- 
tion alone  the  pole  of  the  earth  would  perform  a  revolution  in  a 
very  small  ellipse,  only  18"  in  diameter,  the  center  being  in  the 
circle  which  the  pole  describes  around  the  pole  of  the  ecliptic ; 
but  the  combined  effects  of  precession  and  nutation  convert  the 
circumference  of  this  circle  into  a  wavy  line.  The  motion  of  the 
equator  occasioned  by  nutation,  causes  it  alternately  to  approach 
to  and  recede  from  the  stars,  and  thus  to  change  their  declinations. 
The  solar  nutation,  depending  on  the  position  of  the  sun  with  re- 
spect to  the  equinoxes,  passes  through  all  its  variations  annually ; 
but  the  lunar  nutation  depending  on  the  position  of  the  moon  with 
respect  to  her  nodes,  varies  through  a  period  of  about  18£  years. 

ABERRATION. 

195.  ABERRATION  is  an  apparent  change  of  place  in  the  stars, 
occasioned  by  the  joint  effects  of  the  motion  of  the  earth  in  its  orbit, 
and  the  progressive  motion  of  light. 

Let  EE'  (Fig.  43,)  represent  a  part  of  the  earth's  orbit,  and  SE 
a  ray  of  light  from  the  star  S.  Take  EC  and  EA  proportional 


MOTION    OF    THE    APSIDES.  109 

to  the  velocity  of  each  respectively ;  com- 
plete the  parallelogram,  and  draw  the  diagonal 
EB.  Since  an  object  always  appears  in  the 
direction  in  which  a  ray  of  light  coming  from 
it,  meets  the  eye,  the  combination  of  the  two 
motions  produces  an  impression  on  the  eye 
exactly  similar  to  that  which  would  have  been 
produced  if  the  eye  had  remained  at  rest  in 
the  point  E,  and  the  particle  of  light  had  come 
down  to  it  in  the  direction  S'E  ;  the  star, 
therefore,  whose  place  is  at  S,  will  appear  to 
the  spectator  at  E  to  be  situated  at  S'.  The 
difference  between  its  true  and  its  apparent  place,  that  is,  the 
angle  SES'  is  the  aberration,  the  magnitude  of  which  is  obtained 
from  the  known  ratio  of  EA  to  EC,  or  the  velocity  of  light  to  that 
of  the  earth  in  its  orbit. 

The  velocity  of  light  is  192,000  miles  per  second,  while  that  of 
the  earth  in  its  orbit  is  about  19  miles  per  second.  Represent- 
ing the  velocity  of  light  by  the  line  EA,  and  that  of  the  earth  by 
AB,  then, 

192,000  :  19:  :Rad.  :  tan.  20."5=the  angle  at  E,  which  is  the 
amount  of  aberration  when  the  direction  of  the  ray  of  light  is  per- 
pendicular to  the  earth's  motion. 

The  effect  of  aberration  upon  the  places  of  the  fixed  stars  is  to 
carry  their  apparent  places  a  little  forward  of  their  real  places  in 
the  direction  of  the  earth's  motion.  The  effect  upon  each  particu- 
lar star  will  be  to  make  it  describe  a  small  ellipse  in  the  heavens, 
having  for  its  center  the  point  in  which  the  star  would  be  seen  if 
the  earth  were  at  rest. 


MOTION    OP    THE    APSIDES. 

196.  The  two  points  of  the  ecliptic  where  the  earth  is  at  the 
greatest  and  least  distances  from  the  sun  respectively,  do  not 
always  maintain  the  same  places  among  the  signs,  but  gradually 
shift  their  positions  from  west  to  east.  If  we  accurately  observe 
the  place  among  the  stars,  where  the  earth  is  at  the  time  of  its 
perihelion  the  present  year,  we  shall  find  that  it  will  not  be  pre- 


110  THE    SUN. 

cisely  at  that  point  the  next  year  when  it  arrives  at  its  perihelion, 
but  about  12"  (ll."66)  to  the  east  of  it.  And  since  the  equinox 
itself,  from  which  longitude  is  reckoned,  moves  in  the  opposite 
direction  50."  1  annually,  the  longitude  of  the  perihelion  increases 
every  year  61."76,  or  a  little  more  than  one  minute.  This  fact 
is  expressed  by  saying  that  the  line  of  the  apsides  of  the  earth's 
orbit  has  a  slow  motion  from  west  to  east.  It  completes  one  entire 
revolution  in  its  own  plane  in  about  100,000  years  (111,149.) 

The  mean  longitude  of  the  perihelion  at  the  commencement  of 
the  present  century  was  99°  30'  5",  and  of  course  in  the  ninth 
degree  of  Cancer,  a  little  'past  the  winter  solstice.  In  the  year 
1248,  the  perihelion  was  at  the  place  of  this  solstice  ;  and  since  the 
increase  of  longitude  is  61. "76  a  year,  hence, 

61."76  :  1  : :  90°  :  5246=the  time  occupied  in  passing  from  the 
first  of  Aries  to  the  solstice.  Hence,  5246—1248=3998,  which  is 
the  time  before  the  Christian  era,  when  the  perigee  was  at  the 
first  of  Aries.  But  this  differs  only  6  years  from  the  time  of  the 
creation  of  the  world,  which  is  fixed  by  chronologists  at  4004 
years  A.  C.  At  the  period  of  the  creation,  therefore,  the  line  of 
the  apsides  of  the  earth's  orbit,  coincided  with  the  line  of  the 
equinoxes. 

197.  The  angular  distance  of  a  body  from  its  aphelion  is  called 
its  Anomaly  ;  and  the  interval  between  the  sun's  passing  the  point 
of  the  ecliptic  corresponding  to  the  earth's  aphelion,  and  return- 
ing to  the  same  point  again,  is  called  the  anomalistic  year.     This 
period  must  be  a  little  longer  than. the  sidereal  year,  since,  in  order 
to  complete  the  anomalistic  revolution,  the  sun  must  traverse  an 
arc  of  11. "66  in  addition  to  360°. 

Now  360°  :  365.256 :  :  ll."66  :  4m.  44s. 

198.  Since  the  points  of  the  annual  orbit,  where  the  sun  is  at 
the  greatest  and  least  distances  from  the  earth,  change  their  posi- 
tion with  respect  to  the  solstices,  a  slow  change  is  occasioned  in 
the  duration  of  the  respective  seasons.     For,  let  the  perihelion 
correspond  to  the  place  of  the  winter  solstice,  as  was  the  case  in 
the  year  1248  ;  then  as  the  sun  moves  more  rapidly  in  that  part 
of  his  orbit,  the  winter  months  will  be  shorter  than  the  summer 


MEAN  AND  TRUE  PLACES  OF  THE  SUN.  Ill 

But,  again,  let  the  perihelion  be  at  the  summer  solstice,  as  it  will 
be  in  the  year  6485*  ;  then  the  sun  will  move  most  rapidly 
through  the  summer  months,  and  the  winters  will  be  longer  than 
the  summers.  At  present  the  perihelion  is  so  near  the  winter 
solstice,  that,  the  year  being  divided  into  summer  and  winter  by 
the  equinoxes,  the  six  winter  months  are  passed  over  between  seven 
and  eight  days  sooner  than  the  summer  months. 

MEAN  AND  TRUE  PLACES  OP  THE  SUN. 

199.  The  Mean  Motion  of  any  body  revolving  in  an  orbit,  is 
that  which  it  would  have  if,  in  the  same  time,  it  revolved  uniformly 
in  a  circle. 

In  surveying  an  irregular  field,  it  is  common  first  to  strike  out 
some  regular  figure,  as  a  square  or  a  parallelogram,  by  running 
long  lines,  and  disregarding  many  small  irregularities  in  the  boun- 
daries of  the  field.  By  this  process,  we  obtain  an  approximation 
to  the  contents  of  the  field,  although  we  have  perhaps  thrown  out 
several  small  portions  which  belong  to  it,  and  included  a  number 
of  others  which  do  not  belong  to  it.  These  being  separately  esti- 
mated and  added  to  or  substracted  from  our  first  computation,  we 
obtain  the  true  area  of  the  field.  In  a  similar  manner,  we  proceed 
in  finding  the  place  of  a  heavenly  body,  which  moves  in  an  orbit 
more  or  less  irregular.  Thus  we  estimate  the  sun's  distance  from 
the  vernal  equinox  for  every  day  of  the  year  at  noon,  on  the 
supposition  that  he  moves  uniformly  in  a  circular  orbit :  this  is 
the  sun's  mean  longitude.  We  then  apply  to  this  result  various 
corrections  for  the  irregularity  of  the  sun's  motions,  and  thus  ob- 
•-ain  the  true  longitude. 

200.  The  corrections  applied  to  the  mean  motions  of  a  heav- 
enly body,  in  order  to  obtain  its  true  place,  are  called  Equations. 
Thus  the  elliptical  form  of  the  earth's  orbit,  the  precession  of  the 
equinoxes,  and  the  nutation  of  the  earth's  axis,  severally  affect 
the  place  of  the  sun  in  his  apparent  orbit,  for  which  equations  are 
applied.     In  a  collection  of  Astronomical  Tables,  a  large  part  of 


«  Biot. 


112  THE   SUN. 

the  whole  are  devoted  to  this  object.  They  give  us  the  amount 
of  the  corrections  to  be  applied  under  all  the  circumstances  and 
constantly  varying  relations  in  which  the  sun,  moon,  and  earth 
are  situated  with  respect  to  each  other.  The  angular  distance  of 
the  earth  or  any  planet  from  its  aphelion,  on  the  supposition  that 
it  moves  uniformly  in  a  circle,  is  called  its  Mean  Anomaly :  its 
actual  distance  at  the  same  moment  in  its  orbit  is  called  its  True 
Anomaly.* 

Thus  in  figure  44,  let  AEB  represent  the  orbit  of  the  earth 
having  the  sun  in  one  of  the  foci  at  S.  Upon  AB  describe  the 
circle  AMB.  Let  E  be  the  place  of  the  earth  in  its  orbit,  and  M 
the  corresponding  place  in  the  circle ;  then  the  angle  MCA  is  the 
mean,  and  ESA  the  true  anomaly.  The  difference  between  the 

Fig.  44. 

M 


mean  and  true  anomaly,  MCA— ESA,  is  called  the  the  Equation  of 
the  Center,  being  that  correction  which  depends  on  the  elliptical 
form  of  the  orbit,  or  on  the  distance  of  the  center  of  attraction 
from  the  center  of  the  figure,  that  is,  on  the  eccentricity  of  the 
orbit.  It  is  much  the  greatest  of  all  the  corrections  used  in  finding 
the  sun's  true  longitude,  amounting,  at  its  maximum,  to  nearly  two 
degrees  (1°  55'  26."8.) 

*  In  some  astronomical  treatises,  the  anomaly  is  reckoned  from  the  perihelion. 


CHAPTER  V. 

Of  THE    MOON LUNAR  GEOGRAPHY* PHASES    OF    THE    MOON HER 

REVOLUTIONS. 

201.  NEXT  to  the  Sun,  the  Moon  naturally  claims  our  attention. 
The  Moon  is  an  attendant  or  satellite  to  the  earth,  around  which 

she  revolves  at  the  distance  of  nearly  240,000  miles.  Her  mean 
horizontal  parallax  being  57'  09",f  consequently,  sin.  57'  09"  : 
semi-diameter  of  the  earth  (3956.2)  : :  rad.  :  238,545.  (Art.  87.) 

The  moon's  apparent  diameter  is  31'  7",  and  her  real  diameter 
2160  miles.  For, 

Rad.  :  238,545:  :sin.  15'  33£"  :  1079.6.  =  moon's  semi-diame- 
ter. (See  Fig.  26,  p.  71.) 

And,  since  spheres  are  as  the  cubes  of  the  diameters,  the  vol- 
ume of  the  moon  is  TV  tnat  °f  tne  earth.  Her  density  is  nearly 
|  (.615)  the  density  of  the  earth,  and  her  mass  (=?V><.615)  is 
about  8*0- 

202.  The  moon  shines   by  reflected  light  borrowed  from  the 
sun,  and  when  full,  exhibits  a  disk  of  silvery  brightness,  diversi- 
fied by  extensive  portions  partially  shaded.     By  the  aid  of  the 
telescope,  we  see  undoubted  signs  of  a  varied  surface,  composed 
of  extensive  tracts  of  level  country,  and  numerous  mountains  and 
valleys. 

203.  The  line  which  separates  the  enlightened  from  the  dark 
portions  of  the  moon's  disk,  is  called  the  Terminator.     (See  Fig.  2. 
Frontispiece.)     As  the  terminator  traverses  the  disk  from  new  to 
full  rnoon,  it  appears  through  the  telescope  exceedingly  broken  in 


*  Selenography  is  a  word  more  appropriate  to  a  description  of  the  raoon,  but  is  not 
perhaps  sufficiently  familiarized  by  use. 

t  Baily's  Astronomical  Tables.  '  '  I 

15 


114  THE   MOON. 

some  parts,  but  smooth  in  others,  indicating  that  some  portions  of  the 
lunar  surface  are  uneven  while  others  are  level.  The  broken  re- 
gions appear  brighter  than  the  smooth  tracts.  The  latter  have 
been  taken  for  seas,  but  it  is  supposed  with  more  probability  that 
they  are  extensive  plains,  since  they  are  still  too  uneven  for  the 
perfect  level  assumed  by  bodies  of  water.  That  there  are  moun- 
tains in  the  moon,  is  known  by  several  distinct  indications.  First, 
when  the  moon  is  increasing,  certain  spots  are  illuminated  sooner 
than  the  neighboring  places,  appearing  like  bright  points  beyond 
the  terminator,  within  the  dark  part  of  the  disk.  (See  Fig.  2. 
Frontispiece.)  Secondly,  after  the  terminator  has  passed  over 
them,  they  project  shadows  upon  the  illuminated  part  of  the  disk, 
always  opposite  to  the  Sun,  corresponding  in  shape  to  the  form  of 
the  mountain,  and  undergoing  changes  in  length  from  night  to 
night,  according  as  the  sun  shines  upon  that  part  of  the  moon 
more  or  less  obliquely.  Many  individual  mountains  rise  to  a  great 
height  in  the  midst  of  plains,  and  there  are  several  very  remarka- 
ble mountainous  groups,  extending  from  a  common  center  in  long 
chains. 

204.  That  there  are  also  valleys  in  the  moon,  is  equally  evident 
The  valleys  are  known  to  be  truly  such,  particularly  by  the  man- 
ner in  which  the  light  of  the  sun  falls  upon  them,  illuminating  the 
part  opposite  to  the  sun  while  the  part  adjacent  is  dark,  as  is  the 
case  when  the  light  of  a  lamp  shines  obliquely  into  a  china  cup. 
These  valleys  are  often  remarkably  regular,  and  some  of  them 
almost  perfect  circles.  In  several  instances,  a  circular  chain  of 
mountains  surrounds  an  extensive  valley,  which  appears  nearly 
level,  except  that  a  sharp  mountain  sometimes  rises  from  the  cen- 
ter. The  best  time  for  observing  these  appearances  is  near  the 
first  quarter  of  the  moon,  when  half  the  disk  is  enlightened  ;* 
but  in  studying  the  lunar  geography,  it  is  expedient  to  observe  the 
moon  every  evening  from  new  to  fall,  or  rather  through  her  en- 
tire series  of  changes. 


*  It  is  earnestly  recommended  to  the  student  of  astronomy,  to  examine  the  moon  re- 
peatedly  with  the  best  telescope  he  can  command,  using  low  powers  at  first,  for  the 
sake  of  a  better  light. 


LUNAR    GEOGRAPHY.  115 

205.  The  various  places  on  the  moon's  disk  have  received  ap- 
propriate names.     The  dusky  regions,  being  formerly  supposed  to 
be  seas,  were  named  accordingly ;  and  other  remarkable  places 
have  each  two  names,  one  derived  from  some  well  known  spot  on 
the  earth,  and  the  other  from  some  distinguished  personage.    Thus 
the  same  bright  spot  on  the  surface  of  the  moon  is  called  Mount 
Sinai  or  Tycho,  and  another  Mount  Etna  or  Copernicus.     The 
names  of  individuals,  however,  are  more  used  than  the  others. 
The  frontispiece  exhibits  the  telescopic  appearance  of  the   full 
moon.     A  few  of  the  most  remarkable  points  have  the  following 
names,  corresponding  to  the  numbers  and  letters  on  the  map.     (See 
Frontispiece.) 

1.  Tycho,  A.  Mare  Humorum, 

2.  Kepler,  B.  Mare  Nubium, 

3.  Copernicus,  C.  Mare  Imbrium, 

4.  Aristarchus,  D.  Mare  Nectaris, 

5.  Helicon,  E.  Mare  Tranquilitatis, 

6.  Eratosthenes,  F.  Mare  Serenitatis, 

7.  Plato,  G.  Mare  Fecunditatis, 

8.  Archimedes,  H.  Mare  Crisium. 

9.  Eudoxus, 
10.  Aristotle, 

206.  The  method  of  estimating  the  height  of  lunar  mountains  is 
as  follows. 

Let  ABO  (Fig.  45.)  be  the  illuminated  hemisphere  of  the  moon, 
SO  a  solar  ray  touching  the  moon  in  O,  a  point  in  the  circle  which 
separates  the  enlightened  from  the  dark  part  of  the  moon.  All  the 
part  ODA  will  be  in  darkness ;  but  if  this  part  contains  a  moun- 
tain MF,  so  elevated  that  its  summit  M  reaches  to  the  solar  ray 
SOM,  the  point  M  will  be  enlightened.  Let  E  be  the  place  of  the 
observer  on  the  earth,  the  moon  being  at  any  elongation  from  the 
sun,  as  measured  by  the  angle  EOS.  Draw  the  lines  EM,  EO, 
and  CM,  C  being  the  center  of  the  moon ;  and  let  FM  be  the 
height  of  the  mountain.  Draw  ON  perpendicular  to  EM.  The 
line  EO  being  known,  and  the  angle  OEM  being  measured  with  a 
micrometer,  the  value  of  ON,  the  projection  of  the  lime  OM,  be- 


116 


THE  MOON. 
Fig.  45. 


ON 


comes  known.     Now  OM=  -  -  ;  and  since  OENis  a  very 

small  angle,  EON  may  be  considered  as  a  right  angle  ;  conse- 


quently,  MON=MOE-90.     Therefore  OM= 


ON 


ON 


cos.  (MOE—  90) 

• 

That  is,  the  distance  between  the  summit 


sin.  MOE  sin.  EOS' 
of  the  mountain  and  the  illuminated  part  of  the  moon's  disk,  is 
equal  to  the  projected  distance  as  measured  by  the  micrometer, 
divided  by  the  sine  of  the  moon's  elongation  from  the  sun. 

Suppose  the  distance  OM=?zCO,  where  n  represents  the  frac- 
tion the  part  OM  is  of  CO  as  determined  by  observation.     Then, 

/.  CM=CO 


/.CM-CO  or  FM=CO  (Vl+n2-l)  =i:-CO,  neglecting  the 

higher  powers  of  n,  which  would  be  of  too  little  value  to  be  worth 
taking  into  the  account.  The  value  of  n  has  been  found  in  one 
case  equal  to  TV,  which  gives  the  height  of  the  mountain  equal  to 
^£j  the  semi-diameter  of  the  moon,  that  is,  3}  miles. 

When  the  moon  is  exactly  at  quadrature,  then  EOM  becomes  a 
right  angle,  and  the  value  of  OM  is  obtained  directly  from  actual 
measurement ;  and  having  CO  and  OM,  we  easily  obtain  CM  and 
of  course  FM. 


LUNAR  GEOGRAPHY.  117 

207.  Schroeter,  a  German  astronomer,  estimated  the  heights  of 
the  lunar  mountains  by  observations  on  their  shadows.     He  made 
them  in  some  cases  as  high  as  ^}¥  of  the  semi-diameter  of  the 
moon,  that  is,  about  5  miles.     The  same  astronomer  also  estimates 
the  depths  of  some  of  the  lunar  valleys  at  more  than  four  miles. 
Hence  it  is  inferred  that  the  moon's  surface  is  more  broken  and 
irregular  than  that  of  the  earth,  its  mountains  being  higher  and  its 
valleys  deeper  in  proportion  to  the  size  of  the  moon  than  those  of 
the  earth. 

208.  Dr.  Her&chel  is  supposed  also  to  have  obtained  decisive 
evidence  of  the  existence   of  volcanoes  in  the  moon,   not   only 
from  the  light  afforded  by  their  fires,  but  also  from  the  formation 
of  new  mountains  by  the  accumulation  of  matter  where  fires  had 
been  seen  to  exist,  and  which  remained  after  the  fires  were  extinct. 

209.  Some  indications  of  an  atmosphere  about  the  moon  have 
been  obtained,  the  most  decisive  of  which  are  derived  from  ap- 
pearances of  twilight,  a  phenomenon  that  implies  the  presence 
of  an  atmosphere.     Similar  indications  have  been  detected,  it  is 
supposed,  in  eclipses  of  the  sun,  denoting  a  transparent  refracting 
medium  encompassing  the  moon.     The  lunar  atmosphere,  how- 
ever, if  any  exists,  is  very  inconsiderable  in  extent  and  density 
compared  with  that  of  the  earth.* 

210.  The  improbability  of  our  ever  identifying  artificial  struc- 
tures in  the  moon  may  be  inferred  from  the  fact  that  a  line  one 
mile  in  length  in  the  moon  subtends  an  angle  at  the  eye  of  only 
about  one  second.     If,  therefore,  works  of  art  were  to  have  a  suf- 
ficient horizontal  extent  to  became  visible,  they  can  hardly  be  sup- 
posed to  attain  the  necessary  elevation,  when  we  reflect  that  the 
height  of  the  great  pyramid  of  Egypt  is  less  than  the  sixth  part  of 
a  mile. 

•SeeEd.Encyc.IL598. 


118  THE    MOON. 

PHASES    OF   THE  MOON. 

211.  The  changes  of  the  moon,  commonly  called  her  Phases, 
arise  from  different  portions  of  her  illuminated  side  being  turned 
towards  the  earth  at  different  times.  When  the  moon  is  first 
seen  after  the  setting  sun,  her  form  is  that  of  a  bright  crescent, 
on  the  side  of  the  disk  next  to  the  sun,  while  the  other  portions 
of  the  disk  shine  with  a  feeble  light,  reflected  to  the  moon  from 
the  earth.  Every  night  we  observe  the  moon  to  be  further  and 
further  eastward  of  the  sun,  and  at  the  same  time  the  crescent 
enlarges,  until,  when  the  moon  has  reached  an  elongation  from 
the  sun  of  90°,  half  her  visible  disk  is  enlightened,  and  she  is 
said  to  be  in  her  first  quarter.  The  terminator,  or  line  which 
separates  the  illuminated  from  the  dark  part  of  the  moon,  is  con- 
vex towards  the  sun  from  the  new  moon  to  the  first  quarter,  and 
the  moon  is  said  to  be  horned.  The  extremities  of  the  crescent 
are  called  cusps.  At  the  first  quarter,  the  terminator  becomes  a 
straight  line,  coinciding  with  a  diameter  of  the  disk ;  but  after 
passing  this  point,  the  terminator  becomes  concave  towards  the 
sun,  bounding  that  side  of  the  moon  by  an  elliptical  curve,  when 
the  moon  is  said  to  be  gibbous.  When  the  moon  arrives  at  the 
distance  of  180°  from  the  sun,  the  entire  circle  is  illuminated, 
and  the  moon  is  full.  She  is  then  in  opposition  to  the  sun,  rising 
about  the  time  the  sun  sets.  For  a  week  after  the  full,  the  moon 
appears  gibbous  again,  until,  having  arrived  within  90°  of  the  sun, 
she  resumes  the  same  form  as  at  the  first  quarter,  being  then  at 
her  third  quarter.  From  this  time  until  new  moon,  she  exhibits 
again  the  form  of  a  crescent  before  the  rising  sun,  until  approach- 
ing her  conjunction  with  the  sun,  her  narrow  thread  of  light  is  lost 
in  the  solar  blaze  ;  and  finally,  at  the  moment  of  passing  the  sun, 
the  dark  side  is  wholly  turned  towards  us  and  for  some  time  we 
lose  sight  of  the  moon. 

The  two  points  in  the  orbit  corresponding  to  new  and  full  moon 
respectively,  are  called  by  the  common  name  of  syzygies ;  those 
which  are  90°  from  the  sun  are  called  quadratures;  and  the 
points  half  way  between  the  syzygies  and  quadratures  are  called 
octants.  The  circle  which  divides  the  enlightened  from  the  unen 
lightened  hemisphere  of  the  moon,  is  called  the  circle  of  illumina 


PHASES.  119 

toon ;  that  wnich  divides  the  hemisphere  that  is  turned  towards 
us  from  the  hemisphere  that  is  turned  from  us,  is  called  the  circle 
of  the  disk. 

212.  As  the  moon  is  an  opake  body  of  a  spherical  figure,  and 
borrows  her  light  from  the  sun,  it  is  obvious  that  that  half  only 
which  is  towards  the  sun  can  be  illuminated.  More  or  less  of 
this  side  is  turned  towards  the  earth,  according  as  the  moon  is  at 
a  greater  or  less  elongation  from  the  sun.  The  reason  of  the  dif- 
ferent phases  will  be  best  understood  from  a  diagram.  Therefore 
let  .T  (Fig.  46,)  represent  the  earth,  and  S  the  sun.  Let  A,  B,  C, 
&c.,  be  successive  positions  of  the  moon.  At  A  the  entire  dark 

Fig.  46. 


side  of  the  moon  being  turned  towards  the  earth,  the  disk  would 
be  wholly  invisible.  At  B,  the  circle  of  the  disk  cuts  off  a  small 
part  of  the  enlightened  hemisphere,  which  appears  in  the  heavens 
at  b,  under  the  form  of  a  crescent.  At  C,  the  first  quarter,  the 
circle  of  the  disk  cuts  off  half  the  enlightened  hemisphere,  and  the 
moon  appears  dichotomized  at  c.  In  like  manner  it  will  be  seen 
that  the  appearances  presented  at  D,  E,  F,  &c.,  must  be  those 
represented  at  d,  e,f. 

REVOLUTIONS    OF   THE    MOON. 

213.   The  moon   revolves   around  the  earth  from  west  to  east, 
the  entire  circuit  of  the  heavens  in  about  27£  days. 


120  THE    MOON. 

The  precise  law  of  the  moon's  motions  in  her  revolution  around 
the  earth,  is  ascertained,  as  in  the  case  of  the  sun,  (Art.  155,)  by 
daily  observations  on  her  meridian  altitude  and  right  ascension. 
Thence  are  deduced  by  calculation  her  latitude  and  longitude, 
from  which  we  find,  that  the  moon  describes  on  the  celestial 
sphere  a  great  circle  of  which  the  earth  is  the  center. 

The  period  of  the  moon's  revolution  from  any  point  in  the 
heavens  round  to  the  same  point  again,  is  called  a  month.  A 
sidereal  month  is  the  time  of  the  moon's  passing  from  any  star, 
until  it  returns  to  the  same  star  again.  A  synodical  month*  is 
the  time  from  one  conjunction  or  new  moon  to  another.  The 
synodical  month  is  about  29£  days,  or  more  exactly,  29d.  12h. 
44m.  2S.8=29.53  days.  The  sidereal  month  is  about  two  days 
shorter,  being  27d.  7h.  43m.  1  ls.5=27.32  days.  As  the  sun  and 
moon  are  both  revolving  in  the  same  direction,  and  the  sun  is 
moving  nearly  a  degree  a  day,  during  the  27  days  of  the  moon's 
revolution,  the  sun  must  have  moved  27°.  Now  since  the  moon 
passes  over  360°  in  27.32  days,  her  daily  motion  must  be  13°  17'. 
It  must  therefore  evidently  take  about  two  days  for  the  moon  to 
overtake  the  sun.  The  difference  between  these  two  periods 
may,  however,  be  determined  with  great  exactness.  The  mid- 
dle of  an  eclipse  of  the  sun  marks  the  exact  moment  of  conjunc- 
tion or  new  moon;  and  by  dividing  the  interval  between  any 
two  solar  eclipses  by  the  number  of  revolutions  of  the  moon,  or 
lunations,  we  obtain  the  precise  period  of  the  synodical  month. 
Suppose,  for  example,  two  eclipses  occur  at  an  interval  of  1,000 
lunations ;  then  the  whole  number  of  days  and  parts  of  a  day 
that  compose  the  interval  divided  by  1,000  will  give  the  exact 
time  of  one  lunation. f  The  time  of  the  synodical  month  being 
ascertained,  the  exact  period  of  the  sidereal  month  may  be  derived 
from  it.  For  the  arc  which  the  moon  describes  in  order  to  come 
into  conjunction  with  the  sun,  exceeds  360°  by  the  space  which 


*  nv  and  oSos,  implying  that  the  two  bodies  come  together. 

t  It  might  at  first  view  seem  necessary  to  know  the  period  of  one  lunation  before 
we  could  know  the  number  of  lunations  in  any  given  interval.  This  period  is  known 
very  nearly  from  the  interval  between  one  new  moon  and  another. 


REVOLUTIONS.  121 

the  sun  has  passed  over  since  the  preceding  conjunction,  that  is, 
in  29.53  days.  Therefore, 

365.24  :  360° ::  29.53  :  29°.l=arc  which  the  moon  must  de- 
scribe more  than  360°  in  order  to  overtake  the  sun.  Hence, 

13°  17'  :  Id. ::29°.l  :  2.21d.=difference  between  the  sidereal 
and  synodical  months;  and  29.53— 2.21=27.32,  the  time  of  the 
sidereal  revolution. 

214.  The  moon's  orbit  is  inclined  to  the   ecliptic  in  an  angle  of 
about  5°  (5°  8'  48").     It  crosses  the  ecliptic  in  two  opposite  points 
called  her  nodes.     The  amount  of  inclination  is  ascertained  by 
observations  on  the  moon's  latitude  when  at  a  maximum,  that 
being  of  course  the  greatest  distance  from  the  ecliptic,  and  there- 
fore equal  to  the  inclination  of  the  two  circles. 

215.  The  moon,  at  the  same  age,  crosses  the  meridian  at  differ- 
ent altitudes  at  different  seasons  of  the  year.     The  full  moon,  for 
example,  will  appear  much  further  in  the  south  when  on  the  meri- 
dian at  one  period  of  the  year  than  at  another.     This  is  owing  to 
the  fact  that  the  moon's  path  is  differently  situated  with  respect  to 
the  horizon,  at  a  given  time  of  night  at  different  seasons  of  the 
year.     By  taking  the  ecliptic  on  an  artificial  globe  to  represent 
the  moon's  path,  (which  is  always  near  it,  Art.  214,)  and  recollect- 
ing that  the  new  moon  is  seen  in  the  same  part  of  the  heavens 
with  the  sun,  and  the  full  moon  in  the  opposite  part  of  the  heavens 
from  the  sun,  we  shall  readily  see  that  in  the  winter  the  new 
moons  must  run  low  because  the  sun  does,  and  for  a  similar  rea- 
son the  full  moons  must  run  high.     It  is  equally  apparent  that,  in 
summer,  when  the  sun  runs  high,  the  new  moons  must  cross  the 
meridian  at  a  high,  and  the  full  moons  at  a  low  altitude.     This 
arrangement  gives  us  a  great  advantage  in  respect  to  the  amount 
of  light  received  from  the  moon ;  since  the  full  moon  is  longest 
above  the  horizon  during  the  long  nights  of  winter,  when  her  pre- 
sence is  most  needed.     This  circumstance  is  especially  favorable  to 
the  inhabitants  of  the  polar  regions,  the  moon,  when  full,  travers- 
ing that  part  of  her  orbit  which  lies  north  of  the  equator,  and  of 
course  above  the  horizon  of  the  north  pole,  and  traversing  the  por- 
tion that  lies  south  of  the  equator,  and  below  the  polar  horizon, 

16 


122  THE   MOON. 

when  new.  During  the  polar  winter,  therefore,  the  moon,  from 
the  first  to  the  last  quarter,  is  commonly  above  the  horizon,  while 
the  sun  is  absent ;  whereas,  during  summer,  while  the  sun  is  pre- 
sent, the  moon  is  above  the  horizon  while  describing  her  first  and 
last  quadrants. 

216.  About  the  time  of  the  autumnal  equinox,  the  mpon  when 
near  the  full,  rises  about  sunset  for  a  number  of  nights  in  succes- 
sion ;  and  as  this  is,  in  England,  the  period  of  harvest,  the  phe- 
nomenon is  called  the  Harvest  Moon.     To  understand  the  reason 
of  this,  since  the  moon  is  never  far  from  the  ecliptic,  we  will 
suppose  her  progress  to  be  in  the  ecliptic.     If  the  moon  moved 
in  the  equator,  then,  since  this  great  circle  is  at  right  angles  to 
the  axis  of  the  earth,  all  parts  of  it,  as  the  earth  revolves,  cut  the 
horizon  at  the  same  constant  angle.     But  the  moon's  orbit,  or 
the  ecliptic,  which  is  here  taken  to  represent  it,  being  oblique 
to  the  equator,  cuts  the  horizon  at  different  angles  in  different 
parts,  as  will  easily  be  seen  by  reference  to  an  artificial  globe. 
When  the  first  of  Aries,  or  vernal  equinox,  is  in  the  eastern  hori- 
zon, it  will  be  seen  that  the  ecliptic,  (and  consequently  the  moon's 
orbit,)  makes  its  least  angle  with  the  horizon.     Now  at  the  au- 
tumnal equinox,  the  sun  being  in  Libra,  the  moon  at  the  full  is  in 
Aries,  and  rises  when  the  sun  sets.     On  the  following  evening, 
although  she  has  advanced  in  her  orbit  about  13°,  (Art.  213,)  yet 
her  progress  being  oblique  to  the  horizon,  and  at  a  small  angle 
with  it,  she  will  be  found  at  this  time  but  a  little  way  below  the 
horizon,  compared  with  the  point  where  she  was  at  sunset  the 
preceding  evening.     She  therefore  rises  but  little  later,  and  so 
for  a  week  only  a  little  later  each  evening  than  she  did  the  pre- 
ceding night. 

217.  The  moon  is  about  ^  nearer  to  us  when  near  the  zenith 
than  when  in  the  horizon; 

The  horizontal  distance  CD  (Fig.  47,)  is  nearly  equal  to  AD= 
AD',  which  is  greater  than  CD'  by  AC,  the  semi-diameter  of  the 
earth =aV  the  distance  of  the  moon.  Accordingly,  the  apparent 
diameter  of  the  moon,  when  actually  measured,  is  about  30" 
(which  equals  about  -gV  °f  tne  whole)  greater  when  in  the  zenith 


REVOLUTIONS. 


123 


than  in  the  horizon.  The  apparent  enlargement  of  the  full  moon 
when  rising,  is  owing  to  the  same  causes  as  that  of  the  sun,  as  ex- 
plained in  article  96. 

Fig.  47. 


218.  The  moon  turns  on  its  axis  in  *he  same  time  in  which  it 
revolves  around  the  earth. 

This  is  known  by  the  moon's  always  keeping  nearly  the  same 
face  towards  us,  as  is  indicated  by  the  telescope,  which  could  not 
happen  unless  her  revolution  on  her  axis  kept  pace  with  her  mo- 
tion in  her  orbit.  Thus,  it  will  be  seen  by  inspecting  figure  31, 
that  the  earth  turns  different  faces  towards  the  sun  at  different 
times  ;  and  if 'a  ball  having  one  hemisphere  white  and  the  other 
black  be  carried  around  a  lamp,  it  will  easily  be  seen  that  it  can- 
not present  the  same  face  constantly  towards  the  lamp  unless  it 
turns  once  on  its  axis  while  performing  its  revolution.  The  same 
thing  will  be  observed  when  a  man  walks  around  a  tree,  keeping 
his  face  constantly  towards  it.  Since  however  the  motion  of  the 
moon  on  its  axis  is  uniform,  while  the  motion  in  its  orbit  is  une- 
qual, the  moon  does  in  fact  reveal  to  us  a  little  sometimes  of  one 
side  and  sometimes  of  the  other.  Thus  when  the  ball  above 
mentioned  is  placed  before  the  eye  with  its  light  side  towards  us, 
or  carrying  it  round,  if  it  is  moved  faster  than  it  is  turned  on  its 
axis,  a  portion  of  the  dark  hemisphere  is  brought  into  view  on 
one  side  ;  or  if  it  is  moved  forward  slower  than  it  is  turned  on 
its  axis,  a  portion  of  the  dark  hemisphere  comes  into  view  on  the 
other  side. 

219.  These  appearances  are  called  the  moon's  librations  in  lon- 
gitude.    The  moon  has  also  a  libration  in  latitude,  so  called,  be- 
cause in  one  part  of  her  revolution,  more  of  the  region  around  one 


124  THE  MOON. 

of  the  poles  comes  into  view,  and  in  another  part  of  the  revolu- 
tion, more  of  the  region  around  the  other  pole  ;  which  gives  the  ap- 
pearance of  a  tilting  motion  to  the  moon's  axis.  This  has  nearly  the 
same  cause  with  that  which  occasions  our  change  of  seasons.  The 
moon's  axis  being  inclined  to  that  of  the  ecliptic,  about  1£  degrees, 
(1°  30'  10".8,)  and  always  remaining  parallel  to  itself,  the  circle 
which  divides  the  visible  from  the  invisible  part  of  the  moon,  will 
pass  in  such  a  way  as  to  throw  sometimes  more  of  one  pole  into 
view  and  sometimes  more  of  the  other,  as  would  be  the  case  with 
the  earth  if  seen  from  the  sun.  (See  Fig.  31.) 

The  moon  exhibits  another  phenomenon  of  this  kind  called 
her  diurnal  libration,  depending  on  the  daily  rotation  of  the 
spectator.  She  turns  the  same  face  towards  the  center  of  the 
earth  only,  whereas  we  view  her  from  the  surface.  When  she  is 
on  the  meridian,  we  see  her  disk  nearly  as  though  we  viewed  it 
from  the  center  of  the  earth,  and  hence  in  this  situation  it  is  sub- 
ject to  little  change  ;  but  when  near  the  horizon,  our  circle  of 
vision  takes  in  more  of  the  upper  limb  than  would  be  presented 
to  a  spectator  at  the  center  of  the  earth.  Hence,  from  this  cause, 
we  see  a  portion  of  one  limb  while  the  moon  is  rising,  which  is 
gradually  lost  sight  of,  and  we  see  a  portion  of  the  opposite  limb 
as  the  moon  declines  towards  the  west.  It  will  be  remarked  that 
neither  of  the  foregoing  changes  implies  any  actual  motion  in  the 
moon,  but  that  each  arises  from  a  change  of  position  in  the  spec- 
tator relative  to  the  moon. 

220.  An  inhabitant  of  the  moon  would  have  but  one  day  and 
one  night  during  the  whole  lunar  month  of  29£  days.  One  of 
its  days,  therefore,  is  equal  to  nearly  15  of  ours.  So  protracted 
an  exposure  to  the  sun's  rays,  especially  in  the  equatorial  regions 
of  the  moon,  must  occasion  an  excessive  accumulation  of  heat ; 
and  so  long  an  absence  of  the  sun  must  occasion  a  corresponding 
degree  of  cold.  Each  day  would  be  a  wearisome  summer ;  each 
night  a  severe  winter.*  A  spectator  on  the  side  of  the  moon 
which  is  opposite  to  us  would  never  see  the  earth ;  but  one  on  the 
side  next  to  us  would  see  the  earth  presenting  a  gradual  succession 

*  Francceur,  Uranog.  p.  91. 


REVOLUTIONS.  125 

of  changes  during  his  long  night  of  360  hours.  Soon  after  the 
earth's  conjunction  with  the  sun,  he  would  have  the  light  of  the 
earth  reflected  to  him,  presenting  at  first  a  crescent,  but  enlarging, 
as  the  earth  approaches  its  opposition,  to  a  great  orb,  13  times  as 
large  as  the  full  moon  appears  to  us,  and  affording  nearly  1 3  times 
as  much  light.  Our  seas,  our  plains,  our  mountains,  our  volcanoes, 
and  our  clouds,  would  produce  very  diversified  appearances,  as 
would  the  various  parts  of  the  earth  brought  successively  into 
view  by  its  diurnal  rotation.  The  earth  while  in  view  to  an  in- 
habitant of  the  moon,  would  remain  immovably  fixed  in  the  same 
part  of  the  heavens.  For  being  unconscious  of  his  own  motion 
around  the  earth,  as  we  are  of  our  motion  around  the  sun,  the 
earth  would  seem  to  revolve  around  his  own  planet  from  west  to 
east ;  but,  meanwhile,  his  rotation  along  ^vith  the  moon  on  her 
axis,  would  cause  the  earth  to  have  an  apparent  motion  westward 
at  the  same  rate.  The  two  motions,  therefore,  would  exactly 
balance  each  other,  and  the  earth  would  appear  all  the  while  at 
rest.  The  earth  is  full  to  the  moon  when  the  latter  is  new  to  us ; 
and  universally  the  two  phases  are  complementary  to  each  other.* 

221.  It  has  been  observed  already,  (Art.  214,)  that  the  moon's 
orbit  crosses  the  ecliptic  in  two  opposite  points  called  the  nodes. 
That  which  the  moon  crosses  from  south  to  north,  is  called  the 
ascending  node ;  that  which  the  moon  crosses  from  north  to  south, 
the  descending  node. 

From  the  manner  in  which  the  figure  representing  the  earth's 
orbit  and  that  of  the  moon,  is  commonly  drawn,  the  learner  is 
sometimes  puzzled  to  see  how  the  orbit  of  the  moon  can  cut  the 
ecliptic  in  two  points  directly  opposite  to  each  other.  But  he  must 
reflect  that  the  lunar  orbit  cuts  the  plane  of  the  ecliptic  and  not 
the  earth's  path  in  that  plane,  although  these  respective  points  are 
projected  upon  that  path  in  the  heavens. 

222.  We  have  thus  far  contemplated  the  revolution  of  the  moon 
around  the  earth  as  though  the  earth  were  at  rest.     But,  in  order 
to  have  just  ideas  respecting  the  moon's  motions,  we  must  recol- 
lect that  the  moon  likewise  revolves  along  with  the  earth  around 

*  Francoeur,  p.  92. 


126  THE   MOON. 

the  sun.  It  is  sometimes  said  that  the  earth  carries  the  moon 
along  with  her  in  her  annual  revolution.  This  language  may 
convey  an  erroneous  idea ;  for  the  moon,  as  well  as  the  earth, 
revolves  around  the  sun  under  the  influence  of  two  forces,  and 
would  continue  her  motion  around  the  sun,  were  the  earth  re- 
moved out  of  the  way.  Indeed,  the  moon  is  attracted  towards 
the  sun  2}  times  more  than  towards  the  earth,*  and  would  aban- 
don the  earth  were  not  the  latter  also  carried  along  with  her  by 
the  same  forces.  So  far  as  the  sun  acts  equally  on  both  bodies, 
their  motion  with  respect  to  each  other  would  not  be  disturbed. 
Because  the  gravity  of  the  moon  towards  the  sun  is  found  to  be 
greater,  at  the  conjunction,  than  her  gravity  towards  the  earth, 
some  have  apprehended  that,  if  the  doctrine  of  universal  gravi- 
tation is  true,  the  moon  ought  necessarily  to  abandon  the  earth. 
In  order  to  understand  the  reason  why  it  does  not  do  thus  we 
must  reflect,  that  when  a  body  is  revolving  in  its  orbit  under  the 
action  of  the  projectile  force  and  gravity,  whatever  diminishes 
the  force  of  gravity  while  that  of  projection  remains  the  same, 
causes  the  body  to  recede  from  the  center;  and  whatever  in 
creases  the  amount  of  gravity  carries  the  body  towards  the  center 
Now,  when  the  moon  is  in  conjunction,  her  gravity  towards  the 
earth  acts  in  opposition  to  that  towards  the  sun,  while  her  velocity 
remains  too  great  to  carry  her,  with  what  force  remains,  in  a 
circle  about  the  sun,  and  she  therefore  recedes  from  the  sun,  and 
commences  her  revolution  around  the  earth.  On  arriving  at  the 
opposition,  the  gravity  of  the  earth  conspires  with  that  of  the  sun, 
and  the  moon's  projectile  force  being  less  than  that  required  to 
make  her  revolve  in  a  circular  orbit,  when  attracted  towards  the 
sun  by  the  sum  of  these  forces,  she  accordingly  begins  to  approach 
the  sun  and  descends  again  to  the  conjunction.! 


*  It  is  shown  by  writers  on  Mechanics,  that  the  forces  with  which  bodies  revolving 
in  circular  orbits  tend  towards  their  centers,  are  as  the  radii  of  their  orbits  divided 
by  the  squares  of  their  periodical  times.  Hence,  supposing  the  orbits  of  the  earth  and 
the  moon  to  be  circular,  (and  their  slight  eccentricity  will  not  much  affect  ihe  re- 
sult,) we  have 

400  1 

G  : 


(365.2*)* 
t  M'Laurin's  Discoveries  of  Newton,  B.  ;v,  ch.  5. 


LUNAR  IRREGULARITIES.  12*7 

223.  The  attraction  of  the  sun,  however,  being  every  where 
greater  than  that  of  the  earth,  the  actual  path  of  the  moon  around 
the  sun  is  every  where  concave  towards  the  latter.  Still  the  el- 
liptical path  of  the  moon  around  the  earth,  is  to  be  conceived  of 
in  the  same  way  as  though  both  bodies  were  at  rest  with  respect 
to  the  sun.  Thus,  while  a  steamboat  is  passing  swiftly  around  an 
island,  and  a  man  is  walking  slowly  around  a  post  in  the  cabin, 
the  line  which  he  describes  in  space  between  the  forward  motion 
of  the  boat  and  his  circular  motion  around  the  post,  may  be  every 
where  concave  towards  the  island,  while  his  path  around  the  post 
will  still  be  the  same  as  though  both  were  at  rest.  A  nail  in  the 
rim  of  a  coach  wheel,  will  turn  around  the  axis  of  the  wheel,  when 
the  coach  has  a  forward  motion  in  the  same  manner  as  when  the 
coach  is  at  rest,  although  the  line  actually  described  by  the  nail 
will  be  the  resultant  of  both  motions,  and  very  different  from 
either. 


CHAPTER  VI. 


LUNAR    IRREGULARITIES. 

224.  WE  have  hitherto  regarded  the  moon  as  describing  a  great 
circle  on  the  face  of  the  sky,  such  being  the  visible  orbit  as  seen 
by  projection.  But,  on  more  exact  investigation,  it  is  found  that 
her  orbit  is  not  a  circle,  and  that  her  motions  are  subject  to  very 
numerous  irregularities.  These  will  be  best  understood  in  con- 
nection with  the  causes  on  which  they  depend.  The  law  of  uni- 
versal gravitation  has  been  applied  with  wonderful  success  to  their 
investigation,  and  its  results  have  conspired  with  those  of  long 
continued  observation,  to  furnish  the  means  of  ascertaining  with 
great  exactness  the  place  of  the  moon  in  the  heavens  at  any  given 
instant  of  time,  past  or  future,  and  thus  to  enable  astronomers  to 
determine  longitudes,  to  calculate  eclipses,  and  to  solve  various 
other  problems  of  the  highest  interest.  A  complete  understand- 
ing of  all  the  irregularities  of  the  moon's  motions,  must  be  sought 


128  THE   MOON. 

for  in  more  extensive  treatises  of  astronomy  than  the  present ;  but 
some  general  acquaintance  with  the  subject,  clear  and  intelligible 
as  far  as  it  goes,  may  be  acquired  by  first  gaining  a  distinct  idea 
of  the  mutual  actions  of  the  sun,  the  moon,  and  the  earth. 

225.  The  irregularities  of  the  moon's  motions,  are  due  chiefly  to 
the  disturbing  influence  of  the  sun,  which  operates  in  two  ways  ;  first, 
by  acting  unequally  on  the  earth  and  moon,  and,  secondly,  by  acting 
obliquely  on  the  moon,  on  account  of  the  inclination  of  her  orbit  to 
the  ecliptic.* 

If  the  sun  acted  equally  on  the  earth  and  moon,  and  always  in 
parallel  lines,  this  action  would  serve  only  to  restrain  them  in  their 
annual  motions  round  the  sun,  and  would  not  affect  their  actions 
on  each  other,  or  their  motions  about  their  common  center  of 
gravity.  In  that  case,  if  they  were  allowed  to  fall  directly  to- 
wards the  sun,  they  would  fall  equally,  and  their  respective  situa- 
tions would  not  be  affected  by  their  descending  equally  towards 
it.  We  might  then  conceive  them  as  in  a  plane,  every  part  of 
which  being  equally  acted  on  by  the  sun,  the  whole  plane  would 
descend  towards  the  sun,  but  the  respective  motions  of  the  earth 
and  the  moon  in  this  plane,  would  be  the  same  as  if  it  were  qui- 
escent. Supposing  then  this  plane  and  all  in  it,  to  have  an  annual 
motion  imprinted  on  it,  it  would  move  regularly  round  the  sun, 
while  the  earth  and  moon  would  move  in  it  with  respect  to  each 
other,  as  if  the  plane  were  at  rest,  without  any  irregularities. 
But  because  the  moon  is  nearer  the  sun  in  one  half  of  her  orbit 
than  the  earth  is,  and  in  the  other  half  of  her  orbit  is  at  a  greater 
distance  than  the  earth  from  the  sun,  while  the  power  of  gravity 
is  always  greater  at  a  less  distance  ;  it  follows,  that  in  one  half  of 
her  orbit  the  moon  is  more  attracted  than  the  earth  towards  the 
sun,  and  in  the  other  half  less  attracted  than  the  earth.  The  ex- 
cess of  the  attraction,  in  the  first  case,  and  the  defect  in  the  second, 
constitutes  a  disturbing  force,  to  which  we  may  add  another, 
namely,  that  arising  from  the  oblique  action  of  the  solar  force, 
since  this  action  is  not  directed  in  parallel  lines,  but  in  lines  that 
meet  in  the  center  of  the  sun. 

*  M'Laurin's  Discoveries  of  Newton,  B.  iv,  ch.  4.    La  Place's  Syst.  du  Monde, 
B.  iv,  ch.  5. 


LUNAR  IRREGULARITIES. 


129 


226.  To  see  the  effects  of  this  process,  let  us  suppose  that  the 
projectile  motions  of  the  earth  and  moon  were  destroyed,  and 
that  they  were  allowed  to  fall  freely  towards  the  sun.     If  the 
moon  was  in  conjunction  with  the  sun,  or  in  that  part  of  her  orbit 
which  is  nearest  to  him,  the  moon  would  be  more  attracted  than 
the  earth,  and  fall  with  greater  velocity  towards  the  sun ;  so  that 
the  distance  of  the  moon  from  the  earth  would  be  increased  in  the 
fall.     If  the  moon  was  in  opposition,  or  in  the  part  of  her  orbit 
which  is  furthest  from  the  sun,  she  would  be  less  attracted  than 
the  earth  by  the  sun,  and  would  fall  with  a  less  velocity  towards 
the  sun,  and  would  be  left  behind ;  so  that  the  distance  of  the 
moon  from  the  earth  would  be  increased  in  this  case  also.     If  the 
moon  was  in  one  of  the  quarters,  then  the  earth  and  moon  being 
both  attracted  towards  the  center  of  the  sun,  they  would  both  de- 
scend directly  towards  that  center,  and  by  approaching  it,  they 
would  necessarily  at  the  same  time  approach  each  other,  and  in 
this  case  their  distance  from  each  other  would  be  diminished. 
Now  whenever  the  action  of  the  sun  would  increase  their  distance, 
if  they  were  allowed  to  fall  towards  the  sun,  then  the  sun's  action, 
by  endeavoring  to  separate  them,  diminishes  their  gravity  to  each 
other ;  whenever  the  sun's  action  would  diminish  the  distance,  then 
it  increases  their  mutual  gravitation.      Hence,  in  the  conjunction 
and  opposition,  that  is,  in  the  syzygies,  their  gravity  towards  each 
other  is  diminished  by  the  action  of  the  sun,  while  in  the  quadra- 
tures it  is  increased.     But  it  must  be  remembered  that  it  is  not 
the  total  action  of  the  sun  on  them  that  disturbs  their  motions, 
but  only  that  part  of  it  which  tends  at  one  time  to  separate  them, 
and  at  another  time  to  bring  them  nearer  together.     The  other 
and  far  greater  part,  has  no  other  effect  than  to  retain  them  in 
their  annual  course  around  the  sun. 

227.  Suppose  the  moon  setting  out  from  the  quarter  that  pre- 
cedes the  conjunction  with  a  velocity  that  would  make  her  de- 
scribe an  exact  circle  round  the  earth,  if  the  sun's  action  had  no 
effect  on  her :  since  her  gravity  is  increased  by  that  action,  she  must 
descend  towards  the  earth  and  move  within  that  circle.     Her  or- 
oit  then  would  be  more  curved  than  it  otherwise  would  have  been; 
because  the  addition  to  her  gravity  will  make  her  fall  further  at 

17 


130 


THE  MOON. 


the  end  of  an  arc  below  the  tangent  drawn  at  the  other  end  of  it. 
Her  motion  will  be  thus  accelerated,  and  it  will  continue  to  be 
accelerated  until  she  arrives  at  the  ensuing  conjunction,  because 
the  direction  of  the  sun's  action  upon  her,  during  that  time,  makes 
an  acute  angle  with  the  direction  of  her  motion.  (See  Fig.  41.) 
At  the  conjunction,  her  gravity  towards  the  earth  being  diminished 
by  the  action  of  the  sun,  her  orbit  will  then  be  less  curved,  and 
she  will  be  carried  further  from  the  earth  as  she  moves  to  the  next 
quarter ;  and  because  the  action  of  the  sun  makes  there  an  obtuse 
angle  with  the  direction  of  her  motion,  she  will  be  retarded  in  the 
same  degree  as  she  was  accelerated  before. 

228.  After  this  general  explanation  of  the  mode  in  which  the 
sun  acts  as  a  disturbing  force  on  the  motions  of  the  moon,  the 
learner  will  be  prepared  to  understand  the  mathematical  develop- 
ment of  the  same  doctrine. 

Therefore,  let  ADBC  (Fig.  48,)  be  the  orbit,  nearly  circular,  in 
which  the  moon  M  revolves  in  the  direction  CADB,  round  the 
earth  E.  Let  S  be  the  sun,  and  let 
SE  the  radius  of  the  earth's  orbit, 
be  taken  to  represent  the  force  with 
which  the  earth  gravitates  to  the  sun. 

Then  (Art.  180,)  -^L:  ^  : :  SE  : 
feUj 


=  the  force  by  which  the  sun 
draws  the  moon  in  the  direction 

Cp3 

MS.  Take  MG=^=,,  and  let  the 

SM2 

parallelogram  KF  be  described, 
having  MG  for  its  diagonal,  and 
having  its  sides  parallel  to  EM  and 
ES.  The  force  MG  may  be  re- 
solved into  two,  MF  and  MK,  of 
which  MF,  directed  towards  E,  the 
center  of  the  earth,  increases  the 
gravity  of  the  moon  to  the  earth,  and  does  not  hinder  the  areas 
described  by  the  radius  vector  from  being  proportional  to  the 


LUNAR  IRREGULARITIES.  1  31 

times.  The  other  force  MK  draws  the  moon  in  the  direction  of 
the  line  joining  the  centers  of  the  sun  and  earth.  It  is,  however, 
only  the  excess  of  this  force,  above  the  force  represented  by  SE, 
or  that  which  draws  the  earth  to  the  sun,  which  disturbs  the  rela- 
tive position  of  the  moon  and  earth.  This  is  evident,  for  if  KM 
were  just  equal  to  ES,  no  disturbance  of  the  moon  relative  to  the 
earth  could  arise  from  it.  If  then  ES  be  taken  from  MK,  the  dif- 
ference HK  is  the  whole  force  in  the  direction  parallel  to  SE,  by 
which  the  sun  disturbs  the  relative  position  of  the  moon  and  earth. 
Now,  if  in  MK,  MN  be  taken  equal  to  HK,  and  if  NO  be  drawn 
perpendicular  to  the  radius  vector  EM  produced,  the  force  MN 
may  be  resolved  into  two,  MO  and  ON,  the  first  lessening  the 
gravity  of  the  moon  to  the  earth  ;  and  the  second,  being  parallel 
to  the  tangent  of  the  moon's  orbit  in  M,  accelerates  the  moon's 
motion  from  C  to  A,  and  retards  it  from  A  to  D,  and  so  alternately 
in  the  other  two  quadrants.  Thus  the  whole  solar  force  directed 
to  the  center  of  the  earth,  is  composed  of  the  two  parts  MF  and 
MO,  which  are  sometimes  opposed  to  one  another,  but  which 
never  affect  the  uniform  description  of  the  areas  about  E.  Near 
the  quadratures  the  force  MO  vanishes,  and  the  force  MF,  which 
increases  the  gravity  of  the  moon  to  the  earth,  coincides  with  CE 
or  DE.  As  the  moon  approaches  the  conjunction  at  A,  the  force 
MO  prevails  over  MF,  and  lessens  the  gravity  of  the  moon  to  the 
earth.  In  the  opposite  point  of  the  orbit,  when  the  moon  is  in  op- 
position at  B,  the  force  with  which  the  sun  draws  the  moon  is  less 
than  that  with  which  the  sun  draws  the  earth,  so  that  the  effect  of 
the  solar  force  is  to  separate  the  moon  and  earth,  or  to  increase 
their  distance ;  that  is,  it  is  the  same  as  if,  conceiving  the  earth 
not  to  be  acted  on,  the  sun's  force  drew  the  moon  in  the  direction 
from  E  to  B.  This  force  is  negative,  therefore,  in  respect  to  the 
force  at  A,  and  the  effect  in  both  cases  is  to  draw  the  moon  from 
the  earth  in  a  direction  perpendicular  to  the  line  of  the  quadra- 
tures. Hence,  the  general  result  is,  that  by  the  disturbing  force 
of  the  sun,  the  gravity  to  the  earth  is  increased  at  the  quadratures, 
and  diminished  at  the  syzygies.  It  is  found  by  calculation  that  the 
average  amount  of  this  disturbing  force  is  ^  of  the  moon's 
gravity  to  the  earth.* 

«  Playfair. 


132  THE   MOON. 

229.  With  these  general  principles  in  view,  we  may  now  pro- 
ceed to  investigate  the  figure  of  the  moon's  orbit,  and  the  irregu- 
larities to  which  the  motions  of  this  body  are  subject. 

230.  The  figure  of  the  moon's  orbit  is  an  ellipse,  having  the  earth 
in  one  of  the  foci. 

The  elliptical  figure  of  the  moon's  orbit,  is  revealed  to  us  by  ob- 
servations on  her  changes  in  apparent  diameter,  and  in  her  hori- 
zontal parallax.  First,  we  may  measure  from  day  to  day  the  ap- 
parent diameter  of  the  moon.  Its  variations  being  inversely  as 
the  distances,  (Art.  163,)  they  give  us  at  once  the  relative  distance 
of  each  point  of  observation  from  the  focus.  Secondly,  the  va- 
riations on  the  moon's  horizontal  parallax,  which  also  are  inversely 
as  the  distances,  (Art.  82,)  lead  to  the  same  results.  Observations 
on  the  angular  velocities,  combined  with  the  changes  in  the  lengths 
of  the  radius  vector,  afford  the  means  of  laying  down  a  plot  of  the 
lunar  orbit,  as  in  the  case  of  the  sun,  represented  in  figure  32. 
The  orbit  is  shown  to  be  nearly  an  ellipse,  because  it  is  found  to 
have  the  properties  of  an  ellipse. 

The  moon's  greatest  and  least  apparent  diameters  are  respectively 
33'.518  and  29'.365,  while  her  corresponding  changes  of  parallax 
are  61 '.4  and  53'.8.  The  two  ratios  ought  to  be  equal,  and  we 
shall  find  such  to  be  the  fact  very  nearly,  as  expressed  by  the  fore- 
going numbers ;  for, 

61.4  :  53.8  :  :  33.518  :  29.369. 

The  greatest  and  least  distances  of  the  moon  from  the  earth, 
derived  from  the  parallaxes,  are  63.8419  and  55.9164,  or  nearly 
64  and  56.  the  radius  of  the  earth  being  taken  for  unity.  Hence, 
taking  the  arithmetical  mean,  which  is  59.879,  we  find  that  the 
mean  distance  of  the  moon  from  the  earth  is  very  nearly  60  times 
the  radius  of  the  earth.  The  point  in  the  moon's  orbit  nearest 
the  earth,  is  called  her  perigee  ;  the  point  furthest  from  the  earth, 
her  apogee. 

The  greatest  and  least  apparent  diameters  of  the  sun  are  re- 
spectively 32.583,  and  31.517,  which  numbers  express  also  the  ratio 
of  the  greatest  and  least  distances  of  the  earth  from  the  sun.  By 
comparing  this  ratio  with  that  of  the  distances  of  the  moon,  it  will 
be  seen  that  the  latter  vary  much  more  than  the  former,  and  con- 


LUNAR  IRREGULARITIES.  133 

sequently  that  the  lunar  orbit  is  much  more  eccentric  than  the  so- 
lar. The  eccentricity  of  the  moon's  orbit  is  in  fact  0.0548,  (the 
semi-major  axis  being  as  usual  taken  for  unity)  =T'j  of  its  mean 
distance  from  the  earth,  while  that  of  the  earth  is  only  .01685=^ 
of  its  mean  distance  from  the  sun. 

231.  The  moon's  nodes  constantly  shift  their  positions  in  the  eclip- 
tic from  east  to  west,  at  the  rate  of  19°  35'  per  annum,  returning  to 
the  same  points  in  18.6  years. 

Suppose  the  great  circle  of  the  ecliptic  marked  out  on  the  face 
of  the  sky  in  a  distinct  line,  and  let  us  observe,  at  any  given  time, 
the  exact  point  where  the  moon  crosses  this  line,  which  we  will 
suppose  to  be  close  to  a  certain  star ;  then,  on  its  next  return  to 
that  part  of  the  heavens,  we  shall  find  that  it  crosses  the  ecliptic 
sensibly  to  the  westward  of  that  star,  and  so  on,  further  and  fur- 
ther to  the  westward  every  time  it  crosses  the  ecliptic  at  either 
node.  This  fact  is  expressed  by  saying  that  the  nodes  retrograde 
on  the  ecliptic,  and  that  the  line  which  joins  them,  or  the  line  of 
the  nodes,  revolves  from  east  to  west. 

232.  This  shifting  of  the  moon's  nodes  implies  that  the  lunar 
orbit  is  not  a  curve  returning  into  itself,  but  that  it  more  resem- 
bles a  spiral  like  the  curve  represented  in  figure  49,  where  abc 
represents  the  ecliptic,  and  ABC  the  Fig- 49- 

lunar  orbit,  having  its  nodes  at  C  and 
E,  instead  of  A  and  a ;  consequently, 
the  nodes  shift  backwards  through 
the  arcs  aC  and  AE.  The  manner 
in  which  this  effect  is  produced  may 
be  thus  explained.  That  part  of  the 
solar  force  which  is  parallel  to  the  line  joining  the  centers  of  the 
sun  and  earth,  (See  Fig.  48,)  is  not  in  the  plane  of  the  moon's 
orbit,  (since  this  is  inclined  to  the  ecliptic  about  5°,)  except  when 
the  sun  itself  is  in  that  plane,  or  when  the  line  of  the  nodes  being 
produced,  passes  through  the  sun.  In  all  other  cases  it  is  oblique 
to  the  plane  of  the  orbit,  and  may  be  resolved  into  two  forces, 
one  of  which  is  at  right  angles  to  that  plane,  and  is  directed  to- 
wards the  ecliptic.  This  force  of  course  draws  the  moon  continu 


134 


THE   MOON. 


ally  towards  the  ecliptic,  or  produces  a  continual  deflection  of  the 
moon  from  the  plane  of  her  own  orbit  towards  that  of  the  earth. 
Hence  the  moon  meets  the  plane  of  the  ecliptic  sooner  than  it 
would  have  done  if  that  force  had  not  acted.  At  every  half  revo- 
Jution,  therefore,  the  point  in  which  the  moon  meets  the  ecliptic, 
shifts  in  a  direction  contrary  to  that  of  the  moon's  motion,  or  con- 
trary to  the  order  of  the  signs.  If  the  earth  and  sun  were  at  rest, 
the  effect  of  the  deflecting  force  just  described,  would  be  to  pro- 
duce a  retrograde  motion  of  the  line  of  the  nodes  till  that  line  was 
brought  to  pass  through  the  sun,  and  of  consequence,  the  plane  of 
the  moon's  orbit  to  do  the  same,  after  which  they  would  both  re- 
main in  their  position,  there  being  no  longer  any  force  tending  to 
produce  change  in  either.  But  the  motion  of  the  earth  carries  the 
line  of  the  nodes  out  of  this  position,  and  produces,  by  that  means, 
its  continual  retrogradation.  The  same  force  produces  a  small 
variation  in  the  inclination  of  the  moon's  orbit,  giving  it  an  alter- 
nate increase  and  decrease  within  very  narrow  limits.*  These 
points  will  be  easily  understood  by  the  aid  of  a  diagram.  There- 
fore, let  MN  (Fig.  50,)  be  the  ecliptic,  ANB  the  orbit  of  the  moon, 
the  moon  being  in  L,  and  N  its  descending  node.  Let  the  disturb- 
ing force  of  the  sun  which  tends  to  bring  it  down  to  the  ecliptic 

Fig.  50. 


be  represented  by  L&,  and  its  velocity  in  its  orbit  by  La.  Under 
the  action  of  these  two  forces,  the  moon  will  describe  the  diago- 
nal Lc  of  the  parallelogram  ba,  and  its  orbit  will  be  changed  from 
AN  to  LN' ;  the  node  N  passes  to  N' ;  and  the  exterior  angle  at  N' 
of  the  triangle  LNN'  being  greater  than  the  interior  and  opposite 

*  Playfair. 


LUNAR  IRREGULARITIES.  135 

angle  at  N,  the  inclination  of  the  orbit  is  increased  at  the  node. 
After  the  moon  has  passed  the  ecliptic  to  the  south  side  to  Z,  the 
disturbing  force  Id  produces  a  new  change  of  the  orbit  N'Ze  to 
N"Z/",  and  the  inclination  is  diminished  as  at  N".  In  general, 
while  the  moon  is  receding  from  one  of  the  nodes,  its  inclination  is 
diminishing :  while  it  is  approaching  a  node,  the  inclination  is  in- 
creasing.* 

233.  The  period  occupied  by  the  sun  in  passing  from  one  of 
the  moon's  nodes  until  it  comes  round  to  the  same  node  again,  is 
called  the  synodical  revolution  of  the  node.     This  period  is  shorter 
than  the  sidereal  year,  being  only  about  346£  days.     For  since 
the  node  shifts  its  place  to  the  westward  19°  35'  per  annum,  the 
sun,  in  his  annual  revolution,  comes  to  it  so  much  before  he  com- 
pletes his  entire  circuit ;  and  since  the  sun  moves  about  a  degree 
a  day,  the  synodical  revolution  of  the  node  is  365—19=346,  or 
more  exactly,  346.619851.     The  time  from  one  new  moon,  or 
from  one  full  moon,  to  another,  is  29.5305887  days.     Now  19 
synodical  revolutions  of  the  nodes  contain  very  nearly  223  of 
these  periods. 

For  346.619851X19=6585.78, 

And  29.5305887X223=6585.32. 

Hence,  if  the  sun  and  moon  were  to  leave  the  moon's  node  toge- 
ther, after  the  sun  had  been  round  to  the  same  node  19  times,  the 
moon  would  have  performed  very  nearly  223  synodical  revolu- 
tions, and  would,  therefore,  at  the  end  of  this  period  meet  at  the 
same  node,  to  repeat  the  same  circuit.  And  since  eclipses  of  the 
sun  and  moon  depend  upon  the  relative  position  of  the  sun,  the 
moon,  and  node,  these  phenomena  are  repeated  in  nearly  the  same 
order,  in  each  of  those  periods.  Hence,  this  period,  consisting  of 
about  18  years  and  10  days,  under  the  name  of  the  Saros,  was 
used  by  the  Chaldeans  and  other  ancient  nations  in  predicting 
eclipses. 

234.  The  Metonic  Cycle  is  not  the  same  with  the  Saros,  but 
consists  of  19  tropical  years.     During  this  period  the  moon  makes 

*  Francceur,  Uranog.  p.  158.— Robison's  Phys.  Astronomy,  Art.  264, 


136  THE  MOON. 

very  nearly  235  synodical  revolutions,  and  hence  the  new  and  full 
moons,  if  reckoned  by  periods  of  19  years,  recur  at  the  same 
dates.  If,  for  example,  a  new  moon  fell  on  the  fiftieth  day  of  one 
cycle,  it  would  also  fall  on  the  fiftieth  day  of  each  succeeding  cycle  ; 
and,  since  the  regulation  of  games,  feasts,  and  fasts,  has  been 
made  very  extensively  according  to  new  or  full  moons,  hence  this 
lunar  cycle  has  been  much  used  both  in  ancient  and  modern 
times.  The  Athenians  adopted  it  433  years  before  the  Christian 
era,  for  the  regulation  of  their  calendar,  and  had  it  inscribed  in 
letters  of  gold  on  the  walls  of  the  temple  of  Minerva.  Hence  the 
term  Golden  Number,  which  denotes  the  year  of  the  lunar  cycle. 

235.  The  line  of  the  apsides  of  the  moon's  orbit  revolves  from 
west  to  east  through  her  whole  orbit  in  about  nine  years. 

If,  in  any  revolution  of  the  moon,  we  should  accurately  mark 
the  place  in  the  heavens  where  the  moon  comes  to  its  perigee, 
(Art.  230,)  we  should  find,  that  at  the  next  revolution,  it  would 
come  to  its  perigee  at  a  point  a  little  further  eastward  than  before, 
and  so  on  at  every  revolution,  until,  after  9  years,  it  would  come 
to  its  perigee  at  nearly  the  same  point  as  at  first.  This  fact  is 
expressed  by  saying  that  the  perigee,  and  of  course  the  apogee, 
revolves,  and  that  the  line  which  joins  these  two  points,  or  the  line 
of  the  apsides,  also  revolves. 

The  place  of  the  perigee  may  be  found  by  observing  when  the 
moon  has  the  greatest  apparent  diameter.  But  as  the  magnitude 
of  the  moon  varies  sJowly  at  this  point,  a  better  method  of  ascer- 
taining the  position  of  the  apsides,  is  to  take  two  points  in  the  or- 
bit where  the  variations  in  apparent  diameter  are  most  rapid,  and 
to  find  where  they  are  equal  on  opposite  sides  of  the  orbit.  The 
middle  point  between  the  two  will  give  the  place  of  the  perigee. 

The  angular  distance  of  the  moon  from  her  perigee  in  any  part 
of  her  revolution,  is  called  the  Moon's  Anomaly. 

236.  The  change  of  place  in  the  apsides  of  the  moon's  orbit, 
like  the  shifting  of  the  nodes,  is  caused  by  the  disturbing  influence 
of  the  sun.     If  when  the  moon  sets  out  from  its  perigee,  it  were 
urged  by  no  other  force  than  that  of  projection,  combined  with  its 
gravitation  towards  the  earth,  it  would  describe  a  symmetrical 


LUNAR  IRREGULARITIES.  137 

curve  (Art.  186,)  coming  to  its  apogee  at  the  distance  of  180°. 
But  as  the  mean  disturbing  force  in  the  direction  of  the  radius 
vector  tends,  on  the  whole,  to  diminish  the  gravitation  of  the 
moon  to  the  earth,  the  portion  of  the  path  described  in  an  instant 
will  be  less  deflected  from  her  tangent,  or  less  curved,  than  if  this 
force  did  not  exist.  Hence  the  path  of  the  moon  will  not  inter- 
sect the  radius  vector  at  right  angles,  that  is,  she  will  not  arrive  at 
her  apogee  until  after  passing  more  than  180°  from  her  perigee, 
by  which  means  these  points  will  constantly  shift  their  positions 
from  west  to  east.*  The  motion  of  the  apsides  is  found  to  be  3° 
1'  20"  for  every  sidereal  revolution  of  the  moon. 

237.  On  account  of  the  greater  eccentricity  of  the  moon's  orbit 
above  that  of  the  sun,  the  Equation  of  the  Center,  or  that  correc- 
tion which  is  applied  to  the  moon's  mean  anomaly  to  find  her  true 
anomaly  (Art.  200,)  is  much  greater  than  that  of  the  sun,  being 
when  greatest  more  than  six  degrees,  (6°  17'  12".7,)  while  that  of 
the  sun  is  less  than  two  degrees,  (1°  55'  26".8.) 

The  irregularities  in  the  motions  of  the  moon  may  be  compared 
to  those  of  the  magnetic  needle.  As  a.  first  approximation,  we  say 
that  the  needle  places  itself  in  a  north  and  south  line.  On  closer 
examination,  however,  we  find  that,  at  different  places,  it  deviates 
more  or  less  from  this  line,  and  we  introduce  the  first  great  cor- 
rection under  the  name  of  the  declination  of  the  needle.  But  ob- 
servation shows  us  that  the  declination  alternately  increases  and 
diminishes  every  day,  and  therefore  we  apply  to  the  decimation 
itself  a  second  correction  for  the  diurnal  variation.  Finally,  we 
ascertain,  from  long  continued  observations,  that  the  diurnal  va- 
riation is  affected  by  the  change  of  seasons,  being  greater  in  sum- 
mer than  in  winter,  and  hence  we  apply  to  the  diurnal  variation  a 
third  correction  for  the  annual  variation. 

In  like  manner,  we  shall  find  the  greater  inequalities  of  the 
moon's  motions  are  themselves  subject  to  subordinate  inequalities, 
which  give  rise  to  smaller  equations,  and  these  to  smaller  still,  to 
the  last  degree  of  refinement. 

238.  Next  to  the  equation  of  the  center,  the  greatest  correction 

*  Playfair. 


138  THE  MOON. 

to  be  applied  to  the  moon's  longitude,  is  that  which  belongs  to  the 
Evection.  The  evection  is  a  change  of  form  in  the  lunar  orbit,  by 
which  its  eccentricity  is  sometimes  increased,  and  sometimes 
diminished.  It  depends  on  the  position  of  the  line  of  the  apsides 
with  respect  to  the  line  of  the  syzygies. 

This  irregularity,  and  its  connexion  with  the  place  of  the  peri- 
gee with  respect  to  the  place  of  conjunction  or  opposition,  was 
known  as  a  fact  to  the  ancient  astronomers,  Hipparchus  and 
Ptolemy  ;  but  its  cause  was  first  explained  by  Newton  in  con- 
formity with  the  law  of  universal  gravitation.  It  was  found,  by 
observation,  that  the  equation  of  the  center  itself  was  subject  to  a 
periodical  variation,  being  greater  than  its  mean,  and  greatest  of 
all  when  the  conjunction  or  opposition  takes  place  at  the  perigee 
or  apogee,  and  least  of  all  when  the  conjunction  or  opposition 
takes  place  at  a  point  half  way  between  the  perigee  and  apogee ; 
or,  in  the  more  common  language  of  astronomers,  the  equation  of 
the  center  is  increased  when  the  line  of-  the  apsides  is  in  syzygy, 
and  diminished  when  that  line  is  in  quadrature.  If,  for  example, 
when  the  line  of  the  apsides  is  in  syzygy,  we  compute  the  moon's 
place  by  deducting  the  equation  of  the  center  from  the  mean 
anomaly  (see  Art.  200,)  seven  days  after  conjunction,  the  compu- 
ted longitude  will  be  greater  than  that  determined  by  actual  obser- 
vation, by  about  80  minutes ;  but  if  the  same  estimate  is  made 
when  the  line  of  the  apsides  is  in  quadrature,  the  computed  longi- 
tude will  be  less  than  the  observed,  by  the  same  quantity.  These 
results  plainly  show  a  connexion  between  the  velocity  of  the 
moon's  motions  and  the  position  of  the  line  of  the  apsides  with 
respect  to  the  line  of  the  syzygies. 

239.  Now  any  cause  which,  at  the  perigee,  should  have  the 
effect  to  increase  the  moon's  gravitation  towards  the  earth  beyond 
its  mean,  and,  at  the  apogee,  to  diminish  the  moon's  gravitation 
towards  the  earth,  would  augment  the  difference  between  the 
gravitation  at  the  perigee  and  at  the  apogee,  and  consequently  in- 
crease the  eccentricity  of  the  orbit.  Again,  any  cause  which  at 
the  perigee  should  have  the  effect  to  lessen  the  moon's  gravitation 
towards  the  earth,  and,  at  the  apogee,  to  increase  it,  would  lessen 
the  difference  between  the  two,  and  consequently  diminish  the 


LUNAR   IRREGULARITIES.  139 

eccentricity  of  the  orbit,  or  bring  it  nearer  to  a  circle.  Let  us 
see  if  the  disturbing  force  of  the  sun  produces  these  effects.  The 
sun's  disturbing  force,  as  we  have  seen  in  article  228,  admits  of 
two  resolutions,  one  in  the  direction  of  the  radius  vector,  (OM, 
Fig.  48,)  the  other  (ON)  in  the  direction  of  a  tangent  to  the  orbit. 
First,  let  AB  be  the  line  of  the  apsides  in  syzygy,  A  being  the  place 
of  the  perigee.  The  sun's  disturbing  force  OM  is  greatest  in  the 
direction  of  the  line  of  the  syzygies  ;  yet  depending  as  it  does  on  the 
unequal  action  of  the  sun  upon  the  earth  and  the  moon,  and  being 
greater  as  their  distance  from  each  other  is  greater,  it  is  at  a  mini- 
mum when  acting  at  the  perigee,  and  at  a  maximum  when  acting  at 
the  apogee.  Hence  its  effect  is  to  draw  away  the  moon  from  the 
earth  less  than  usual  at  the  perigee,  and  of  course  to  make  her 
gravitation  towards  the  earth  greater  than  usual,  while  at  the 
apogee  its  effect  is  to  diminish  the  tendency  of  the  moon  to  the 
earth  more  than  usual,  and  thus  to  increase  the  disproportion  be- 
tween the  two  distances  of  the  moon  from  the  focus  at  these  two 
points,  and  of  course  to  increase  the  eccentricity  of  the  orbit. 
The  moon,  therefore,  if  moving  towards  the  perigee,  is  brought 
to  the  line  of  the  apsides  in  a  point  between  its  mean  place  and 
the  earth  ;  or  if  moving  towards  the  apogee,  she  reaches  the  line 
of  the  apsides  in  a  point  more  remote  from  the  earth  than  its  mean 
place. 

Secondly,  let  CD  be  the  line  of  the  apsides,  in  quadrature,  C 
being  the  place  of  the  perigee.  The  effect  of  the  sun's  disturb- 
ing force  is  to  increase  the  tendency  of  the  moon  towards  the 
earth  when  in  quadrature.  If,  however,  the  moon  is  then  at  her 
perigee,  such  increase  will  be  less  than  usual,  and  if  at  her  apogee, 
it  will  be  more  than  usual ;  hence  its  effect  will  be  to  lessen  the 
disproportion  between  the  two  distances  of  the  moon  from  the 
focus  at  these  two  points ;  and  of  course  to  diminish  the  eccen- 
tricity of  the  orbit.  The  moon,  therefore,  if  moving  towards 
the  perigee,  meets  the  line  of  the  apsides  in  a  point  more  remote 
from  the  earth  than  the  mean  place  of  the  perigee  ;  and  if  moving 
towards  the  apogee,  in  a  point  between  the  earth  and  the  mean  place 
of  the  apogee.* 

*  Woodhouse's  Ast.  p.  680. 


140  THE    MOON. 

240.  A  third  inequality  in  the  lunar  motions,  is  the  Variation. 
By  comparing  the  moon's  place  as  computed  from  her  mean  mo- 
tion corrected  for  the  equation  of  the  center  and  for  evection, 
with  her  place  as  determined  by  observation,  Tycho  Brahe  dis- 
covered that  the  computed  and  observed  places  did  not  always 
agree.     They  agreed  only  in  the  syzygies  and  quadratures,  and 
disagreed  most  at  a  point  half  way  between  these,  that  is,  at  the 
octants.     Here,  at  the  maximum,  it  amounted  to  more  than  half 
a  degree  (35'  41. "6.)     It  appeared  evident  from  examining  the 
daily  observations  while  the  moon  is  performing  her  revolution 
around  the  earth,  that  this  inequality  is  connected  with  the  angular 
distance  of  the  moon  from  the  sun,  and  in  every  part  of  the  orbit 
could  be  correctly  expressed  by  multiplying  the  maximum  value 
as  given  above,  into  the  sine  of  twice  the  angular  distance  between 
the  sun  and  the  moon.     It  is,  therefore,  0  at  the  conjunctions  and 
quadratures,  and  greatest  at  the  octants.     Tycho  Brahe  knew  the 
fact :  Newton  investigated  the  cause. 

It  appears  by  article  228,  that  the  sun's  disturbing  force  can  be 
resolved  into  two  parts, — one  in  the  direction  of  the  radius  vector, 
the  other  at  right  angles  to  it  in  the  direction  of  a  tangent  to  the 
moon's  orbit.  The  former,  as  already  explained,  produces  the 
Evection:  the  latter  produces  the  Variation.  This  latter  force 
will  accelerate  the  moon's  velocity,  in  every  point  of  the  quadrant 
which  the  moon  describes  in  moving  from  quadrature  to  conjunc- 
tion, or  from  C  to  A,  (Fig.  48,)  but  at  an  unequal  rate,  the 
acceleration  being  greatest  at  the  octant,  and  nothing  at  the  quad- 
rature and  the  conjunction ;  and  when  the  moon  is  past  conjunction, 
the  tangential  force  will  change  its  direction  and  retard  the  moon's 
motion.  All  these  points  will  be  understood  by  inspection  of 
figure  48. 

241.  A  fourth  lunar  inequality  is  the  Annual  Equation.     This 
depends  on  the  distance  of  the  earth  (and  of  course  the  moon) 
from  the  sun.     Since  the  disturbing  influence  of  the  sun  has  a 
greater  effect  in  proportion  as  the  sun  is  nearer,*  consequently  all 
the  inequalities  depending  on  this  influence  must  vary  at  different 

*  Varying  reciprocally  as  the  cube  of  the  sun's  distance  from  the  earth. 


LUNAR   IRREGULARITIES.  141 

seasons  of  the  year.     Hence,  the  amount  of  this  effect  due  to  any 
particular  time  of  the  year  is  called  the  Annual  Equation. 

242.  The  foregoing  are  the  largest  of  the  inequalities  of  the 
moon's  motions,  and  may  serve  as  specimens  of  the  corrections  that 
are  to  be  applied  to  the  mean  place  of  the  moon  in  order  to  find 
her  true  place.     These  were  first  discovered  by  actual  observa- 
tion ;  but  a  far  greater  number,  though  most  of  them  are  exceed- 
ingly minute,  have  been  made  known  by  the  investigations  of  Phys- 
ical Astronomy,  in  following  out  all  the  consequences  of  universal 
gravitation.     In  the  best  tables,  about  30  equations  are  applied  to 
the  mean  motions  of  the  moon.     That  is,  we  first  compute  the 
place  of  the  moon  on  the  supposition  that  she  moves  uniformly 
in  a  circle.     This  gives  us  her  mean  place.     We  then  proceed, 
by  the  aid  of  the  Lunar  Tables,  to  apply  the  different  corrections, 
such  as  the  equation  of  the  center,  evection,  variation,  the  annual 
equation,  and  so  on,  to  the  number  of  28.     Numerous  as  these 
corrections  appear,  yet  La  Place  informs  us,  that  the  whole  num- 
ber belonging  to  the  moon's  longitude  is  no  less  than  60 ;  and 
that  to  give  the  tables  all  the  requisite  degree  of  precision,  addi- 
tional investigations  will  be  necessary,  as  extensive  at  least  as 
those  already  made.*     The  best  tables  in  use  in  the  time  of  Tycho 
Brahe,  gave  the  moon's  place  only  by  a  distant  approximation. 
The  tables  in  use  in  the  time  of  Newton,  (Halley's  tables,)  approxi- 
mated within  7  minutes.     Tables  at  present  in  use  give  the  moon's 
place  to  5  seconds.     These  additional  degrees  of  accuracy  have 
been  attained  only  by  immense  labor,  and  by  the  united  efforts  of 
Physical  Astronomy  and  the  most  refined  observations. 

243.  The  inequalities  of  the  moon's  motions  are  divided  into 
periodical  and  secular.     Periodical  inequalities  are  those  which 
are  completed  in  comparatively  short  periods,  like  evection  and 
variation:   Secular   inequalities  are   those  which  are  completed 
only  in  very  long  periods,  such  as  centuries  or  ages.     Hence  the 
corresponding  terms  periodical  equations,  and  secular  equations. 
As  an  example  of  a  secular  inequality,  we  may  mention  the  ac~ 

*  Syst.  du  Monde,  1.  iv,c.  5. 


142  THE  MOON. 

celeration  of  the  moon's  mean  motion.  It  is  discovered,  that  the 
moon  actually  revolves  around  the  earth  in  less  time  now  than 
she  did  in  ancient  times.  The  difference  however  is  exceedingly 
small,  being  only  about  10"  in  a  century,  but  increases  from  century 
to  century  as  the  square  of  the  number  of  centuries  from  a  given 
epoch.  This  remarkable  fact  was  discovered  by  Dr.  Halley.*  In  a 
lunar  eclipse  the  moon's  longitude  differs  from  that  of  the  sun,  at  the 
middle  of  the  eclipse,  by  exactly  180° ;  and  since  the  sun's  lon- 
gitude at  any  given  time  of  the  year  is  known,  if  we  can  learn 
the  day  and  hour  when  an  eclipse  occurs,  we  shall  of  course  know 
the  longitude  of  the  sun  and  moon.  Now  in  the  year  721  before 
the  Christian  era,  on  a  specified  day  and  hour,  Ptolemy  records  a 
lunar  eclipse  to  have  happened,  and  to  have  been  observed  by 
the  Chaldeans.  The  moon's  longitude,  therefore,  for  that  time  is 
known  ;  and  as  we  know  the  mean  motions  of  the  moon  at  pre- 
sent, starting  from  that  epoch,  and  computing,  as  may  easily  be 
done,  the  place  which  the  moon  ought  to  occupy  at  present  at  any 
given  time,  she  is  found  to  be  actually  nearly  a  degree  and  a  half 
in  advance  of  that  place.  Moreover,  the  same  conclusion  is 
derived  from  a  comparison  of  the  Chaldean  observations  with  those 
made  by  an  Arabian  astronomer  of  the  tenth  century. 

This  phenomenon  at  first  led  astronomers  to  apprehend  that  the 
moon  encountered  a  resisting  medium,  which,  by  destroying  at 
every  revolution  a  small  portion  of  her  projectile  force,  would 
have  the  effect  to  bring  her  nearer  and  nearer  to  the  earth  and 
thus  to  augment  her  velocity.  But  in  1786,  La  Place  demon- 
strated that  this  acceleration  is  one  of  the  legitimate  effects  of  the 
sun's  disturbing  force,  and  is  so  connected  with  changes  in  the 
eccentricity  of  the  earth's  orbit,  that  the  moon  will  continue  to  be 
accelerated  while  that  eccentricity  diminishes,  but  when  the  eccen- 
tricity has  reached  its  minimum  (as  it  will  do  after  many  ages) 
and  begins  to  increase,  then  the  moon's  motion  will  begin  to  be 
retarded,  and  thus  her  mean  motions  will  oscillate  forever  about  a 
mean  value. 

244.  The  lunar  inequalities  which  have  been  considered  are  such 

*  Astronomer  Royal  of  Great  Britain,  and  cotemporary  with  Sir  Isaac  Newton. 


ECLIPSES.  143 

only  as  affect  the  moon's  longitude  ;  but  the  sun's  disturbing  force 
also  causes  inequalities  in  the  moon's  latitude  and  parallax.  Those 
of  latitude  alone  require  no  less  than  twelve  equations.  Since 
the  moon  revolves  in  an  orbit  inclined  to  the  ecliptic,  it  is  easy  to 
see  that  the  oblique  action  of  the  sun  must  admit  of  a  resolution 
into  two  forces,  one  of  which  being  perpendicular  to  the  moon's 
orbit,  must  effect  changes  in  her  latitude.  Since  also  several  of  the 
inequalities  already  noticed  involve  changes  in  the  length  of  the 
radius  vector,  it  is  obvious  that  the  moon's  parallax  must  be  sub- 
ject to  corresponding  perturbations. 


CHAPTER    VII. 


ECLIPSES. 


245.  AN  eclipse  of  the  moon  happens,  when  the  moon  in  its 
revolution  about  the  earth,  falls  into  the  earth's  shadow.  An 
eclipse  of  the  sun  happens,  when  the  moon,  coming  between  the 
earth  and  the  sun,  covers  either  a  part  or  the  whole  of  the  solar 
disk.  An  eclipse  of  the  sun  can  occur  only  at  the  time  of  con- 
junction, or  new  moon ;  and  an  eclipse  of  the  moon,  only  at  the 
time  of  opposition,  or  full  moon.  Were  the  moon's  orbit  in  the 
same  plane  with  that  of  the  earth,  or  did  it  coincide  with  the 
ecliptic,  then  an  eclipse  of  the  sun  would  take  place  at  every 
conjunction,  and  an  eclipse  of  the  moon  at  every  opposition ;  for 
as  the  sun  and  earth  both  lie  in  the  ecliptic,  the  shadow  of  the 
earth  must  also  extend  in  the  same  plane,  being  of  course  always 
directly  opposite  to  the  sun  ;  and  since,  as  we  shall  soon  see,  the 
length  of  this  shadow  is  much  greater  than  the  distance  of  the 
moon  from  the  earth,  the  moon,  if  it  revolved  in  the  plane  of  the 
ecliptic,  must  pass  through  the  shadow  at  every  full  moon.  For 
similar  reasons,  the  moon  would  occasion  an  eclipse  of  the  sun, 
partial  or  total,  in  some  portions  of  the  earth  at  every  new  moon. 
But  the  lunar  orbit  is  inclined  to  the  ecliptic  about  5°,  so  that  the 
center  of  the  moon,  when  she  is  furthest  from  her  node,  is  5°  from 


144  THE  MOON. 

the  axis  of  the  earth's  shadow  (which  is  always  in  the  ecliptic ;) 
and,  as  we  shall  show  presently,  the  greatest  distance  to  which  the 
shadow  extends  on  each  side  of  the  ecliptic,  that  is,  the  greatest 
semi-diameter  of  the  shadow,  where  the  moon  passes  through  it, 
is  only  about  £  of  a  degree,  while  the  semi-diameter  of  the  moon's 
disk  is  only  about  j  of  a  degree  ;  hence  the  two  semi-diame- 
ters, namely,  that  of  the  moon  and  the  earth's  shadow,  cannot 
overlap  one  another,  unless,  at  the  time  of  new  or  full  moon,  the 
sun  is  at  or  very  near  the  moon's  node.  In  the  course  of  the  sun's 
apparent  revolution  around  the  earth  once  a  year,  he  is  succes- 
sively in  every  part  of  the  ecliptic  ;  consequently,  the  conjunctions 
and  oppositions  of  the  sun  and  moon  may  occur  at  any  part  of  the 
ecliptic,  either  when  the  sun  is  at  the  moon's  node,  (or  when  he 
is  passing  that  point  of  the  celestial  vault  on  which  the  moon's 
node  is  projected  as  seen  from  the  earth  ;)  or  they  may  occur 
when  the  sun  is  90°  from  the  moon's  node,  where  the  lunar  and 
solar  orbits  are  at  the  greatest  distance  from  each  other;  or,  finally, 
they  may  occur  at  any  intermediate  point.  Now  the  sun,  in  his 
annual  revolution,  passes  each  of  the  moon's  nodes  on  opposite 
sides  of  the  ecliptic,  and  of  course  at  opposite  seasons  of  the 
year  ;  so  that,  for  any  given  year,  the  eclipses  commonly  happen 
in  two  opposite  months,  as  January  and  July,  February  and 
August,  May  and  November.  These,  therefore,  are  called  Node 
Months. 

246.  If  the  sun  were  of  the  same  size  with  the  earth,  the  shadow 
of  the  earth  would  be  cylindrical  and  infinite  in  length,  since  the 
tangents  drawn  from  the  sun  to  the  earth  (which  form  the  bounda- 
ries of  the  shadow)  would  be  parallel  to  each  other ;  but  as  the 
sun  is  a  vastly  larger  body  than  the  earth,  the  tangents  converge 
and  meet  in  a  point  at  some  distance  behind  the  earth,  forming  a 
cone  of  which  the  earth  is  the  base,  and  whose  vertex  (and  of 
course  its  axis)  lies  in  the  ecliptic.  A  little  reflection  will  also 
show  us,  that  the  form  and  dimensions  of  the  shadow  must  be 
affected  by  several  circumstances ;  that  the  shadow  must  be  of 
the  greatest  length  and  breadth  when  the  sun  is  furthest  from  the 
earth  ;  that  its  figure  will  be  slightly  modified  by  the  spheroidal 
figure  of  the  earth  ;  and  that  the  moon,  being,  at  the  time  of  i* 


ECLIPSES.  145 

opposition,  sometimes  nearer  to  the  earth,  and  sometimes  further 
from  it,  will  accordingly  traverse  it  at  points  where  its  breadth 
varies  more  or  less. 

247.   The  semi-angle  of  the  cone  of  the  earth's  shadow,  is  equal 
to   the  sun's   apparent   semi-diameter,   minus  his  horizontal  pat 
allax, 

Let  AS  (Fig.  51,)  be  the  semi-diameter  of  the  sun,  BE  that  of 
the  earth,  and  EC  the  axis  of  the  earth's  shadow.  Then  the 
semi-angle  of  the  cone  of  the  earth's  shadow  ECB=AES-EAB, 

Fig.  51. 
A 


of  which  AES  is  the  sun's  semi-diameter  and  EAB  his  horizontal 
parallax  ;  and  as  both  these  quantities  are  known,  hence  the  angle 
at  the  vertex  of  the  shadow  becomes  known.  Putting  <5  for  the 
the  sun's  semi-diameter,  andp  for  his  horizontal  parallax,  we  have 
the  semi-angle  of  the  earth's  shadow  ECB=£— p. 

248.  At  the  mean  distance  of  the  earth  from  the  sun,  the  length 
of  the  earth's  shadow  is  about  860,000  miles,  or  more  than  three  times 
the  distance  of  the  moon  from  the  earth. 

In  the  right  angled  triangle  ECB,  right  angled  at  B,  the  angle 
ECB  being  known,  and  the  side  EB,  we  can  find  the  side  EC. 

FT5 

For  sin.  (5— p)  :  EB::R  :  EC=-^--J- — .     This  value  will  vary 

sin.  (d— p) 

with  the  sun's  semi-diameter,  being  greater  as  that  is  less.  Its 
mean  value  being  16'  1".5  and  the  sun's  horizontal  parallax  being 
9".6,  5—p=I5r  52".9,  and  EB=3956.2.  Hence, 

Sin.  15'  53"  :  Rad. : :  3956.2  :  856,275. 

.  Since  the  distance  of  the  moon  from  the  earth  is  238,545  miles, 
the  shadow  extends  about  3.6  times  as  far  as  the  moon,  and  con- 

19 


146  THE  MOON. 

sequently,  the  moon  passes  the  shadow  towards  its  broadest  part, 
where  its  breadth  is  much  more  than  sufficient  to  cover  the  moon's 
disk. 

249.  The  average  breadth  of  the  earth1  s  shadow  where  it  eclipses 
the  moon  is  almost  three  times  the  moon's  diameter. 

Let  mm'  (Fig.  51,)  represent  a  section  of  the  earth's  shadow 
where  the  moon  passes  through  it,  M  being  the  center  of  the  cir- 
cular section.  Then  the  angle  MEm  will  be  the  angular  breadth 
of  half  the  shadow.  But, 

MEm  =  BwE  —  BCE  ;  that  is,  since  ~BmE  is  the  moon's  horizon- 
tal parallax,  (Art.  82,)  and  BCE  equals  the  sun's  semi-diameter 
minus  his  horizontal  parallax  (&—p,)  therefore,  putting  P  for  the 
moon's  horizontal  parallax,  we  have 

MEm  =  'P-(d-p)=P+p-5'J  that  is,  since  P=57'  1"  and 
S-p=l5>  52".9,  therefore,  57'  I"— 15'  52".9=41'  8".l,  which  is 
nearly  three  times  15'  33",  the  semi-diameter  of  the  moon.  Thus, 
it  is  seen  how,  by  the  aid  of  geometry,  we  learn  to  estimate  vari- 
ous particulars  respecting  the  earth's  shadow,  by  means  of  simple 
data  derived  from  observation. 

250.  The  distance  of  the  moon  from  her  node  when  she  just 
touches  the  shadow  of  the  earth,  in  a  lunar  eclipse,  is  called  the 
Lunar  Ecliptic  Limit ;  and  her  distance  from  the  node  in  a  solar 
eclipse,  when  the  moon  just  touches  the  solar  disk,  is  called  the 
Solar  Ecliptic  Limit.     The  Limits  are  respectively  the  furthest 
possible  distances  from  the  node  at  which  eclipses  can  take  place. 

251.  The  Lunar  Ecliptic  Limit  is  nearly  12  degrees. 

Let  CN  (Fig.  52,)  be  the  sun's  path,  MN  the  moon's,  and  N  the 
node.  Let  Ca  be  the  semi-diameter  of  the  earth's  shadow,  and 
Ma  the  semi-diameter  of  the  moon.  Since  Ca  and  Ma  are  known 

Fig.  52. 


ECLIPSES.  147 

quantities,  their  sum  CM  is  also  known.  The  angle  at  N  is 
known,  being  the  inclination  of  the  lunar  orbit  to  the  ecliptic. 
Hence,  in  the  spherical  triangle  MCN,  right  angled  at  M,*  by 
Napier's  theorem,  (Art.  132,  Note,) 

Rad.xsin.  CM=sin.  CNxsin.  MNC. 

The  greatest  apparent  semi-diameter  of  the  earth's  shadow 
where  the  moon  crosses  it,  computed  by  article  249,  is  45'  52", 
and  the  moon's  greatest  apparent  semi-diameter,  is  16'  45".5, 
which  together,  give  MC  equal  to  62'  37". 5.  Taking  the  incli- 
nation of  the  moon's  orbit,  or  the  angle  MNC  (what  it  generally 
is  in  these  circumstances)  at  5°  17',  and  we  have  Rad.xsin. 

62'  37".5=sin.  CNxsin.  5°  17',  or  sin.  CN-Rad- ^^f7''-8. 

and  CN^ll0  25'  40".  f  This  is  the  greatest  distance  of  the  moon  from 
her  node  at  which  an  eclipse  of  the  moon  can  take  place.  By 
varying  the  value  of  CM,  corresponding  to  variations  in  the  dis- 
tances of  the  sun  and  moon  from  the  earth,  it  is  found  that  if  NC 
is  less  than  9°,  there  must  be  an  eclipse  ;  but  between  this  and  the 
limit,  the  case  is  doubtful. 

When  the  moon's  disk  only  comes  in  contact  with  the  earth's 
shadow,  as  in  figure  52 ,  the  phenomenon  is  called  an  appulse, 
when  only  a  part  of  the  disk  enters  the  shadow,  the  eclipse  is 
said  to  be  partial,  and  Mai  if  the  whole  of  the  disk  enters  the 
the  shadow.  The  eclipse  is  called  central  when  the  moon's  center 
coincides  with  the  axis  of  the  shadow,  which  happens  when  the. 
moon  at  the  time  of  the  eclipse  is  exactly  at  her  node. 

252.  Before  the  moon  enters  the  earth's  shadow,  the  earth  be- 
gins to  intercept  from  it  portions  of  the  sun's  light,  gradually  in- 
creasing until  the  moon  reaches  the  shadow.  This  partial  light  is 
called  the  moon's  Penumbra.  Its  limits  are  ascertained  by  drawing 
the  tangents  AC'B'  and  A'C'B.  (Fig.  51.)  Throughout  the  space 
included  between  these  tangents  more  or  less  of  the  sun's  light  is 
intercepted  from  the  moon  by  the  interposition  of  the  earth ;  for 

*  The  line  CM  is  to  be  regarded  as  the  projection  of  the  line  which  connects  the 
centers  of  the  moon  and  section  of  the  earth's  shadow,  as  seen  from  the  earth, 
t  Woodhouse's  Astronomy,  p.  718. 


148  THE    MOON. 

it  is  evident,  that  as  the  moon  moves  towards  the  shadow,  she 
would  gradually  lose  the  view  of  the  sun,  until,  on  entering  the 
shadow,  the  sun  would  be  entirely  hidden  from  her. 

253.  The  semi-angle  of  the  Penumbra  equals  the  sun's  semi- 
diameter  and  horizontal  parallax,  or  5+p. 

The  angle  7*C'M  (Fig.  51,)=AC'S=AES+B'AE.  But  AES  is 
the  sun's  semi-diameter,  and  B'AE  is  the  sun's  horizontal  parallax, 
both  of  which  quantities  are  known. 

254.  The  semi-angle  of  a  section  of  the  Penumbra,  where  the 
moon  crosses  it,  equals  the  moon's  horizontal  parallax,  plus  the  sun's, 
plus  the  sun's  semi-diameter. 

The  angle  hEM  (Fig.  51,)  =EhC'+EC'h.  But  EhC'=V,  the 
moon's  horizontal  parallax,  and  EC'^  =8-\-p  (Art.  253,)  .*.  7iEM 
=P+p+<5,  all  which  are  likewise  known  quantities. 

Hence,  by  means  of  these  few  elements,  which  are  known  from 
observation,  we  ascend  to  a  complete  knowledge  of  all  the  par- 
ticulars necessary  to  be  known  respecting  the  moon's  penumbra. 

255.  In  the  preceding  investigations,  we  have  supposed  that 
the  cone  of  the  earth's  shadow  is  formed  by  lines  drawn  from  the 
sun,  and  touching  the  earth's  surface.     But  the  apparent  diameter 
of  the  shadow  is  found  by  observation  to  be  somewhat  greater  than 
would  result  from  this  hypothesis.     The  fact  is  accounted  for  by 
supposing  that  a  portion  of  the  solar  rays  which  graze  the  earth's 
surface  are  absorbed  and  extinguished  by  the  lower  strata  of  the 
atmosphere.     This  amounts  to  the  same  thing  as  though  the  earth 
were  larger  than  it  is,  in  which  case  the  moon's  horizontal  parallax 
would  be  increased ;  and  accordingly,  in  order  that  theory  and 
observation  may  coincide,  it  is  found  necessary  to  increase  the 
parallax  by  gV 

256.  In  a  total  eclipse   of  the   moon,  its  disk  is  still  visible, 
shining  with  a  dull  red  light.     This  light  cannot  be  derived  di- 
rectly from  the  sun,  since  the  view  of  the  sun  is  completely  hid- 
den from  the  moon ;  nor  by  reflexion  from  the  earth,  since  the 
illuminated  side  of  the  earth  is  wholly  turned  from  the  moon ;  but 


ECLIPSES.  149 

it  is  owing  to  refraction  by  the  earth's  atmosphere,  by  which  a  few 
scattered  rays  of  the  sun  are  bent  round  into  the  earth's  shadow 
and  conveyed  to  the  moon,  sufficient  in  number  to  afford  the  feeble 
light  in  question. 

257.  In  calculating  an  eclipse  of  the  moon,  we  first  learn  from 
the  tables  in  what  month  the  sun,  at  the  time  of  full  moon  in  that 
month,  is  near  the  moon's  node,  or  within  the  lunar  ecliptic  limit. 
This  it  must  evidently  be  easy  to  determine,  since  the  tables  ena- 
ble us  to  find  both  the  longitudes  of  the  nodes,  and  the  longitudes 
of  the  sun  and  moon,  for  every  day  of  the  year.     Consequently, 
we  can  find  when  the  sun  has  nearly  the  same  longitude  as  one  of 
the  nodes,  and  also  the  precise  moment  when  the  longitude  of  the 
moon  is  180°  from  that  of  the  sun,  for  this  is  the  time  of  opposition, 
from  which  may  be  derived  the  time  of  the  middle  of  the  eclipse. 
Having  the  time  of  the  middle  of  the  eclipse,  and  the  breadth 
of  the  shadow,  (Art.  249,)  and  knowing,  from  the  tables,  the  rate 
at  which  the  moon  moves  per  hour  faster  than  the  shadow,  we  can 
find  how  long  it  will  take  her  to  traverse  half  the  breadth  of  the 
shadow ;  and  this  time  subtracted  from  the  time  of  the  middle 
of  the  eclipse,  will  give  the  beginning,  and  added  to  the  time  of 
the  middle  will  give  the  end  of  the  eclipse.     Or  if  instead  of  the 
breadth  of  the  shadow,  we  employ  the  breadth  of  the  penumbra 
(Art.  253,)  we  may  find,  in  the  same  manner,  when  the  moon 
enters  and  when  she  leaves  the  penumbra.     We  see,  therefore, 
how  by  having  a  few  things  known  by  observation,  such  as  the 
sun  and  moon's  semi-diameters,  and  their  horizontal  parallaxes, 
we  rise,  by  the  aid  of  trigonometry,  to  the  knowledge  of  various 
particulars  respecting  the  length  and  breadth  of  the  shadow  and 
of  the  penumbra.     These  being  known,  we  next  have  recourse  to 
the  tables  which  contain  all  the  necessary  particulars  respecting 
the  motions  of  the  sun  and  moon,  together  with  equations  or  cor- 
rections, to  be  applied  for  all  their  irregularities.     Hence  it  is  com- 
paratively an  easy  task  to  calculate  with  great  accuracy  an  eclipse 
of  the  moon. 

258.  Let  us  then  see  how  we  may  find  the  exact  time  of  the  be- 
ginning, end,  duration,  and  magnitude,  of  a  lunar  eclipse. 


150  THE   MOON. 

Let  NG  (Fig  53,)  be  the  ecliptic,  and  "Nag  the  moon's  orbit,  the 
sun  being  in  A*  when  the  moon  is  in  opposition  at  a ;  let  N  be 
the  ascending  node,  and  Aa  the  moon's  latitude  at  the  instant 

Fig.  53. 


of  opposition.  An  hour  afterwards  the  sun  will  have  passed  to 
A',  and  the  moon  to  g,  when  the  difference  of  longitude  of  the  two 
bodies  will  be  GA'.  Then  gh  is  the  moon's  hourly  motion  in  lati- 
tude, and  ah  her  hourly  motion  in  longitude.  As  the  character 
and  form  of  the  eclipse  will  depend  solely  upon  the  distances 
between  the  centers  of  the  sun  and  moon,  that  is,  upon  the  line 
gA',  instead  of  considering  the  two  bodies  as  both  in  motion, 
we  may  suppose  the  sun  at  rest  in  A,  and  the  moon  as  advancing 
with  a  motion  equal  to  the  difference  between  its  rate  and  that 
of  the  sun,  a  supposition  which  will  simplify  the  calculation. 
Therefore,  draw  gd  parallel  and  equal  to  A'A,  join  dA,  and  this 
line  being  equal  to  gA',  the  two  bodies  will  be  in  the  same  relative 
situation  as  if  the  sun  were  at  A'  and  the  moon  at  g.  Join  da  and 
produce  the  line  da  both  ways,  cutting  the  ecliptic  in  F;  then 
da¥  will  be  the  moon's  Relative  Orbit.  Hence  ai—dh— AA'=the 
difference  of  the  hourly  motions  of  the  sun  and  moon,  that  is,  the 
moon's  relative  motion  in  longitude,  and  di=ihe  moon's  hourly 
motion  in  latitude. 

Draw  CD  (Fig.  54,)  to  represent  the  ecliptic,  and  let  A  be  the 
place  of  the  sun.  As  the  tables  give  the  computation  of  the 
moon's  latitude  at  every  instant,  consequently,  we  may  take  from 
them  the  latitude  corresponding  to  the  instant  of  opposition,  and 
to  one  hour  later ;  and  we  may  take  also  the  sun's  and  moon's 
hourly  motions  in  longitude.  Take  AD,  AB,  each  equal  to  fhe 
relative  motion,  and  A«=the  latitude  in  opposition,  Dd=the  lati- 


*  It  will  be  remarked  that  the  point  A  really  represents  the  center  of  the  earth's 
shadow  ;  but  as  the  real  motions  of  the  shadow  are  the  same  with  the  assumed  motions 
of  the  sun,  the  latter  are  used  in  conformity  with  the  language  of  the  tables. 


ECLIPSES.  151 

Fig.  54. 


B  C 


tude  one  hour  afterwards ;  join  da  and  produce  the  line  da  both 
ways,  and  it  will  represent  the  moon's  relative  orbit.  Draw  B6 
at  right  angles  to  CD,  and  it  will  be  the  latitude  an  hour  before 
opposition.  At  the  time  of  the  eclipse,  the  apparent  distance  of 
the  center  of  the  shadow  from  the  moon  is  very  small ;  conse- 
quently, CD,  cd,  DC?,  &c.  may  be  regarded  as  straight  lines. 
During  the  short  interval  between  the  beginning  and  end  of  an 
eclipse,  the  motion  of  the  sun,  and  consequently  that  of  the  cen- 
ter of  the  shadow,  may  likewise  be  regarded  as  uniform. 

259.  The  various  particulars  that  enter  into  the  calculation  of 
an  eclipse  are  called  its  Elements  ;  and  as  our  object  is  here  merely 
to  explain  the  method  of  calculating  an  eclipse  of  the  moon,  (refer- 
ring to  the  Supplement  for  the  actual  computation,)  we  may  take 
the  elements  at  their  mean  value.  Thus,  we  will  consider  cd  as 
inclined  to  CD  5°  9',  the  moon's  horizontal  parallax  as  58',  its  semi- 
diameter  as  16',  and  that  of  the  earth's  shadow  as  42'.  The  line 
Am  perpendicular  to  cd  gives  the  point  m  for  the  place  of  the 
moon  at  the  middle  of  the  eclipse,  for  this  line  bisects  the  chord, 
which  represents  the  path  of  the  moon  through  the  shadow ;  and 
mM.,  perpendicular  to  CD,  gives  AM  for  the  time  of  the  middle 
of  the  eclipse  before  opposition,  the  number  of  minutes  before  op- 
position being  the  same  part  of  an  hour  that  AM  is  of  AB.*  From 
the  center  A,  with  a  radius  equal  to  that  of  the  earth's  shadow 
(42')  describe  the  semi-circle  BLF,  and  it  will  represent  the  pro- 
jection of  the  shadow  traversed  by  the  moon.  With  a  radius 
equal  to  the  semi-diameter  of  the  shadow  and  that  of  the  moon 


*  The  situation  of  the  moon  when  at  m  is  called  orbit  opposition ;  and  her  situation 
when  at  a,  ecliptic  opposition. 


152  THE  MOON. 

(= 42'+ 16' =58')  and  with  the  center  A,  mark  the  two  points  c  and 
f  on  the  relative  orbit,  and  they  will  be  the  places  of  the  center 
of  the  moon  at  the  beginning  and  end  of  the  eclipse.  The  per- 
pendiculars cC,/F,  give  the  times  AC  and  AF  of  the  commence- 
ment and  the  end  of  the  eclipse,  and  CM,  or  MF  gives  half  the 
duration.  From  the  centers  c  and  f  with  a  radius  equal  to  the 
semi-diameter  of  the  moon  (16')  describe  circles,  and  they  will 
each  touch  the  shadow,  (Euc.  3.12.)  indicating  the  position  of  the 
moon  at  the  beginning  and  end  of  the  eclipse.  If  the  same  circle 
described  from  m  is  wholly  within  the  shadow,  the  eclipse  will  be 
total;  if  it  is  only  partly  within  the  shadow,  the  eclipse  will  be 
partial.  With  the  center  A,  and  radius  equal  to  the  semi-diame- 
ter of  the  shadow  minus  that  of  the  moon  (42'— 16' =26')  mark 
the  two  points  c',f,  which  will  give  the  places  of  the  center  of  the 
moon,  at  the  beginning  and  end  of  total  darkness,  and  MC',  MF' 
will  give  the  corresponding  times  before  and  after  the  middle  of 
the  eclipse.  Their  sum  will  be  the  duration  of  total  darkness. 

260.  If  the  foregoing  projection  be  accurately  made  from  a  scale, 
the  required  particulars  of  the  eclipse  may  be  ascertained  by 
measuring  on  the  same  scale,  the  lines  which  respectively  repre- 
sent them ;  and  we  should  thus  obtain  a  near  approximation  to  the 
elements  of  the  eclipse.  A  more  accurate  determination  of  these 
elements  may,  however,  be  obtained  by  actual  calculation.  The 
general  principles  of  the  calculation  will  be  readily  understood. 

First,  knowing  ai,  (Fig.  53,)  the  moon's  relative  longitude,  and 
di,  her  latitude,  we  find  the  angle  dai,  which  is  the  inclination  of 
the  moon's  relative  orbit.  But  dai=aAm ;  and,  in  the  triangle 
aAm,  we  have  the  angle  at  A,  and  the  side  A#,  being  the  moon's 
latitude  at  the  time  of  opposition,  which  is  given  by  the  tables. 
Hence  we  can  find  the  side  Am.  In  the  triangle  AmM,  (Fig.  54,) 
having  the  side  Am  and  the  angle  AmM.  (=aAm)  we  can  find  AM 
=  the  arc  of  relative  longitude  described  by  the  moon  from  the 
time  of  the  middle  of  the  eclipse  to  the  time  of  opposition ;  and 
knowing  the  moon's  hourly  motion  in  longitude,  we  can  convert 
AM  into  time,  and  this  subtracted  from  the  time  of  opposition 
gives  us  the  time  of  the  middle  of  the  eclipse. 


ECLIPSES.  153 

Secondly,  since  we  know  the  length  of  the  line  Ac*  (Fig.  54) 
and  can  easily  find  the  angle  cAC,  we  can  thus  obtain  the  side 
AC  ;  and  AC—  AM  =MC,  which  arc,  converted  into  time  by  com- 
paring it  with  the  moon's  hourly  motion  in  longitude,  gives  us, 
when  subtracted  from  the  time  of  the  middle  of  the  eclipse,  the 
time  of  the  beginning  of  the  eclipse,  or  when  added  to  that  of  the 
middle,  the  time  of  the  end  of  the  eclipse.  The  sum  of  the  two 
equals  the  whole  duration. 

Thirdly,  by  a  similar  method  we  calculate  the  value  of  MC', 
which  converted  into  time,  and  subtracted  from  the  time  of  the 
middle  of  the  eclipse,  gives  the  commencement  of  total  darkness,  or 
when  added  gives  the  end  of  total  darkness.  Their  sum  is  the 
duration  of  total  darkness. 

Fourthly,  the  quantity  of  the  eclipse  is  determined  by  supposing 
the  diameter  of  the  moon  divided  into  twelve  equal  parts  called 
Digits,  and  finding-  how  many  such  parts  lie  within  the  shadow, 
at  the  time  when  the  centers  of  the  moon  and  the  shadow  are 
nearest  to  each  other.  Even  when  the  moon  lies  wholly  within 
the  shadow,  the  quantity  of  the  eclipse  is  still  expressed  by  the  num- 
ber of  digits  contained  in  that  part  of  the  line  which  joins  the  cen- 
ter of  the  'shadow  and  the  center  of  the  moon,  which  is  intercepted 
between  the  edge  of  the  shadow  and  the  inner  edge  of  the  moon. 

Thus  in  figure  54,  the  number  of  digits  eclipsed,  equals  -  - 

T17M 

_    o—    n_   o—(    m—nm)  anexpresg^on  containing  only  known 


quantities. 

261.  The  foregoing  will  serve  as  an  explanation  of  the  generaT 
principles,  on  which  proceeds  the  calculation  of  a  lunar  eclipse. 
The  actual  methods  practiced  employ  many  expedients  to  facili- 
tate the  process,  and  to  insure  the  greatest  possible  accuracy,  the 
nature  of  which  are  explained  and  exemplified  in  Mason's  Supple- 
ment to  this  work. 

262.  The   leading  particulars    respecting  an  ECLIPSE  or  THE 
SUN,  are  ascertained  very  nearly  like  those  of  a  lunar  eclipse.   The 

*  This  line  is  not  represented  in  the  figure,  but  may  be  easily  imagined. 

20 


154  THE  MOON. 

shadow  of  the  moon  travels  over  a  portion  of  the  earth,  as  the 
shadow  of  a  small  cloud,  seen  from  an  eminence  in  a  clear  day, 
rides  along  over  hills  and  plains.  Let  us  imagine  ourselves  stand- 
ing on  the  moon ;  then  we  shall  see  the  earth  partially  eclipsed  by 
the  shadow  of  the  moon,  in  the  same  manner  as  we  now  see  the 
moon  eclipsed  by  the  earth's  shadow  ;  and  we  might  proceed  to 
find  the  length  of  the  shadow,  its  breadth  where  it  eclipses  the 
earth,  the  breadth  of  the  penumbra,  and  its  duration  and  quantity, 
in  the  same  way  as  we  have  ascertained  these  particulars  for  an 
eclipse  of  the  moon. 

But,  although  the  general  characters  of  a  solar  eclipse  might  be 
investigated  on  these  principles,  so  far  as  respects  the  earth  at 
large,  yet  as  the  appearances  of  the  same  eclipse  of  the  sun  are 
very  different  at  different  places  on  the  earth's  surface,  it  is  neces- 
sary to  calculate  its  peculiar  aspects  for  each  place  separately,  a 
circumstance  which  makes  the  calculation  of  a  solar  eclipse  much 
more  complicated  and  tedious  than  of  an  eclipse  of  the  moon. 
The  moon,  when  she  enters  the  shadow  of  the  earth,  is  deprived 
of  the  light  of  the  part  immersed,  and  that  part  appears  black 
alike  to  all  places  where  the  moon  is  above  the  horizon.  But  it  is 
not  so  with  a  solar  eclipse.  We  do  not  see  this  by  the  shadow 
cast  on  the  earth,  as  we  should  do  if  we  stood  on  the  moon,  but 
by  the  interposition  of  the  moon  between  us  and  the  sun  ;  and  the 
sun  may  be  hidden  from  one  observer  while  he  is  in  full  view  of 
another  only  a  few  miles  distant.  Thus,  a  small  insulated  cloud 
sailing  in  a  clear  sky,  will,  for  a  few  moments,  hide  the  sun  from 
us,  and  from  a  certain  space  near  us,  while  all  the  region  around 
is  illuminated. 

263.  We  have  compared  the  motion  of  the  moon's  shadow  over 
the  surface  of  the  earth  to  that  of  a  cloud ;  but  its  velocity  is  in- 
comparably greater.  The  mean  motion  of  the  moon  around  the 
earth  is  about  33'  per  hour ;  but  33'  of  the  moon's  orbit  is  2280 
miles,  and  the  shadow  moves  of  course  at  the  same  rate,  or  2280 
miles  per  hour,  traversing  the  entire  disk  of  the  earth  in  less  than 
four  hours.  This  is  the  velocity  of  the  shadow  when  it  passes 
perpendicularly  over  the  earth ;  when  the  direction  of  the  axis  of 
the  shadow  is  oblique  to  the  earth's  surface,  the  velocity  is  increased 


ECLIPSES.  1 55 

in  proportion  of  radius  to  the  sine  of  obliquity.  Thus  the  shadows 
of  evening  have  a  far  more  rapid  motion  than  those  of  noon-day. 
Let  us  endeavor  to  form  a  just  conception  of  the  manner  in 
which  these  three  bodies,  the  sun,  the  earth,  and  the  moon,  are 
situated  with  respect  to  each  other  at  the  time  of  a  solar  eclipse. 
First,  suppose  the  conjunction  to  take  place  at  the  node.  Then 
the  straight  line  which  connects  the  centers  of  the  sun  and  the 
earth,  also  passes  through  the  center  of  the  moon,  and  coincides 
with  the  axis  of  its  shadow  ;  and,  since  the  earth  is  bisected  by 
the  plane  of  the  ecliptic,  the  shadow  would  traverse  the  earth  in 
the  direction  of  the  terrestrial  ecliptic,  from  west  to  east,  passing 
over  the  middle  regions  of  the  earth.  Here  the  diurnal  motion  of 
the  earth  being  in  the  same  direction  with  the  shadow,  but  with  a 
less  velocity,  the  shadow  will  appear  to  move  with  a  speed  equal 
only  to  the  difference  between  the  two.  Secondly,  suppose  the 
moon  is  on  the  north  side  of  the  ecliptic  at  the  time  of  conjunction, 
and  moving  towards  her  descending  node,  and  that  the  conjunc- 
tion takes  place  just  within  the  solar  ecliptic  limit,  say  16°  from  the 
node.  The  shadow  will  now  not  fall  in  the  plane  of  the  ecliptic, 
but  a  little  northward  of  it,  so  as  just  to  graze  the  earth  near  the 
pole  of  the  ecliptic.  The  nearer  the  conjunction  comes  to  the 
node,  the  further  the  shadow  will  fall  from  the  pole  of  the  ecliptic 
towards  the  equatorial  regions.  In  certain  cases,  the  shadow 
strikes  beyond  the  pole  of  the  earth ;  and  then  its  easterly  motion 
being  opposite  to  the  diurnal  motion  of  the  places  which  it  traver- 
ses, consequently  its  velocity  is  greatly  increased,  being  equal  to 
the  sum  of  both. 

264.  After  these  general  considerations,  we  will  now  examine 
more  particularly  the  method  of  investigating  the  elements  of  a 
solar  eclipse. 

The  length  of  the  moon's  shadow,  is  the  first  object  of  inquiry. 
The  moon,  as  well  as  the  earth,  is  at  different  distances  from  the 
sun  at  different  times,  and  hence  the  length  of  her  shadow  varies, 
being  always  greatest  when  she  is  furthest  from  the  sun.  Also, 
since  her  distance  from  the  earth  varies,  the  section  of  the  moon's 
shadow  made  by  the  earth,  is  greater  in  proportion  as  the  moon  is 


156  THE  MOON. 

nearer  the  earth.     The  greatest  eclipses  of  the  sun,  therefore, 
happen  when  the  sun  is  in  apogee,*  and  the  moon  in  perigee. 

265.  WJien  the  moon  is  at  her  mean  distance  from  the  earth,  and 
from  the  sun,  her  shadow  nearly  reaches  the  earth's  surface. 

Let  S  (Fig.  55,)  represent  the  sun,  D  the  moon,  and  T  the 
earth.  Then,  the  semi-angle  of  the  cone  of  the  moon's  shadow, 
DKR,  will,  as  in  the  case  of  the  earth,  (Art.  247,)  equal  SDR— 
DRK,  of  which  SDR  is  the  sun's  apparent  semi-diameter,  as  seen 
from  the  moon,  and  DRK,  is  the  sun's  horizontal  parallax  at  the 
moon.  Since,  on  account  of  the  great  distance  of  the  sun,  corn- 
Fig.  55. 


pared  with  that  of  the  moon,  the  semi-diameter  of  the  sun  as  seen 
from  the  moon,  must  evidently  be  very  nearly  the  same  as 
when  seen  from  the  earth,  and  since  on  account  of  the  minute- 
ness of  the  moon's  semi-diameter  when  seen  from  the  sun,  the 
sun's  horizontal  parallax  at  the  moon  must  be  very  small,  we  might, 
without  much  error,  take  the  siin's  apparent  semi-diameter  from 
the  earth,  as  equal  to  the  semi-angle  of  the  cone  of  the  moon's 
shadow ;  but,  for  the  sake  of  greater  accuracy,  let  us  estimate  the 
value  of  the  sun's  semi-diameter  and  horizontal  parallax  at  the 
moon. 

Now,  SDR  :  STR  :  :  ST  :  SDf  :  :  400  :  399  ;   hence   SDR  = 

—  STR=1.0025  STR  ;  and  the  sun's  mean  semi-diameter  STR 
399 

being  16.025,  hence  SDR=1.0025xl6.025=16.065=16'  3".9. 

Again,  since  parallax  is  inversely  as  the  distance,  the  sun's  hor- 
izontal parallax  at  the  moon,  is  on  account  of  her  being  nearer  the 
sun  ^  greater  than  at  the  earth ;  but  on  account  of  her  inferior 

*  The  sun  is  said  to  be  in  apogee,  when  the  earth  is  in  aphelion, 
t  The  apparent  magnitude  of  an  object  being  reciprocally  as  its  distance  from  the 
eye.    See  Note,  p.  86. 


ECLIPSES.  157 

size  it  is  |f  j£  less  than  at  the  earth.     Hence,  increasing  the  sun's 
horizontal  parallax  at  the  earth  by  the  former  fraction,  and  dimin- 

ishing it  by  the  latter,  we  have-—  x  —  ^-x9"=2".5=the  sun's 


horizontal  parallax  at  the  moon.  Therefore,  the  semi-angle  of  the 
cone  of  the  moon's  shadow,  which,  as  appears  above,  equals 
SDR—  DRK,  equals  16'  3".9-2".5=16'  1".4,  which  so  nearly 
equals  the  sun's  apparent  semi-diameter,  as  seen  from  the  earth, 
that  we  may  adopt  the  latter  as  the  value  of  the  semi-angle  of  the 
shadow.  Hence,  sin.  16'  1".5  :  1080  (BD)  :  :  Rad.  :  DK=231690. 
But  the  mean  distance  of  the  moon  from  the  surface  of  the  earth 
is  238545  —  3956=234589,  which  exceeds  a  little  the  mean  length 
of  the  shadow  as  above. 

But  when  the  moon  is  nearest  the  earth  her  distance  from  the 
center  of  the  earth  is  only  221148  miles;  and  when  the  earth  is 
furthest  from  the  sun,  the  sun's  apparent  semi-diameter  is  only 
15'  45".5.  By  employing  this  number  in  the  foregoing  estimate, 
we  shall  find  the  length  of  the  shadow  235630  miles;  and 
235630—221148=14482,  the  distance  which  the  moon's  shadow 
may  reach  beyond  the  center  of  the  earth.  . 

266.  The  diameter  of  the  moon's  shadow  where  it  traverses  the 
earth,  is,  at  its  maximum,  about  170  miles.* 

In  the  triangle  eTK,  the  angle  at  K=15'  45".5  (Art.  265,)  the 
side  Te=3956,  and  TK=14482. 

Or,  3956  :  14482  :  :  sin.  15'  45".5  :  sin.  57'  41".5. 

And  57'  41".5+15'  45".5=1°  13'  21"=dTe,  or  the  arc  de. 

And  2de=2°  26'  54"=en. 

Hence  360  :  2.45  (=2°  26'  54")  :  :  24899f  :  170  (nearly). 

267.  The  greatest  portion  of  the  earths  surface  ever  covered  by 
the  moon's  penumbra,  is  about  4393  miles. 

The  semi-angle  of  the  penumbra  BID=BSD+SBR,  of  which 
BSD  the  sun's  horizontal  parallax  at  the  moon  =2".  5,  and  SBR 
the  sun's  apparent  semi-diameter  =16'  3".9,  and  hence  BID  is 

*  This  supposes  the  conjunction  to  take  place  at  the  node,  and  the  shadow  to  strike 
the  earth  perpendicularly  to  its  surface  ;  where  it  strikes  obliquely,  the  section  may  be 
greater  than  this. 

t  The  equatorial  circumference. 


158  THE  MOON. 

known.  The  moon's  apparent  semi-diameter  BCD =16'  45'  .5. 
Therefore  GDT  is  known,  as  likewise  DT  and  TG.  Hence  the 
angle  GTd  may  be  found,  and  the  arc  dG  and  its  double  GH, 
which  equals  the  angular  breadth  of  the  penumbra.  It  may  be 
converted  into  miles  by  stating  a  proportion  as  in  article  266. 
On  making  the  calculation  it  will  be  found  to  be  4393  miles. 

268.  The  apparent  diameter  of  the  moon  is  sometimes  larger 
than  that  of  the  sun,  sometimes  smaller,  and  sometimes  exactly 
equal  to  it.  Suppose  an  observer  placed  on  the  right  line  which 
joins  the  centers  of  the  sun  and  moon  ;  if  the  apparent  diameter  of 
the  moon  is  greater  than  that  of  the  sun,  the  eclipse  will  be  total. 
If  the  two  diameters  are  equal,  the  moon's  shadow  just  reaches  the 
earth,  and  the  sun  is  hidden  but  for  a  moment  from  the  view  of 
spectators  situated  in  the  line  which  the  vertex  of  the  shadow  de- 
scribes on  the  surface  of  the  earth.  But  if,  as  happens  when  the 
moon  comes  to  her  conjunction  in  that  part  of  her  orbit  which  is 
towards  her  apogee,  the  moon's  diameter  is  less  than  the  sun's, 
then  the  observer  will  see  a  ring  of  the  sun  encircle  the  moon, 
constituting  an  annular  eclipse.  (Fig.  55'.) 

Fig.  55'. 


269.  Eclipses  of  the  sun  are  modified  by  the  elevation  of  the 
moon  above  the  horizon,  since  its  apparent  diameter  is  augmented 


ECLIPSES.  159 

as  its  altitude  is  increased,  (Art.  217.)  This  effect,  combined  with 
that  of  parallax,  may  so  increase  or  diminish  the  apparent  distance 
between  the  centers  of  the  sun  and  moon,  that  from  this  cause 
alone,  of  two  observers  at  a  distance  from  each  other,  one  might 
see  an  eclipse  which  was  not  visible  to  the  other.*  If  the  hori- 
zontal diameter  of  the  moon  differs  but  little  from  the  apparent 
diameter  of  the  sun,  the  case  might  occur  where  the  eclipse  would 
be  annular  over  the  places  where  it  was  observed  morning  and 
evening,  but  total  where  it  was  observed  at  mid-day. 

The  earth  in  its  diurnal  revolution  and  the  moon's  shadow  both 
move  from  west  to  east,  but  the  shadow  moves  faster  than  the 
earth ;  hence  the  moon  overtakes  the  sun  on  its  western  limb  and 
crosses  it  from  west  to  east.  The  excess  of  the  apparent  diame- 
ter of  the  moon  above  that  of  the  sun  in  a  total  eclipse  is  so  small, 
that  total  darkness  seldom  continues  longer  than  four  minutes,  and 
can  never  continue  so  long  as  eight  minutes.  An  annular  eclipse 
may  last  12m.  24s. 

Since  the  sun's  ecliptic  limits  are  more  than  17°  and  the  moon's 
less  than  12°,  eclipses  of  the  sun  are  more  frequent  than  those  of 
the  moon.  Yet  lunar  eclipses  being  visible  to  every  part  of  the 
terrestrial  hemisphere  opposite  to  the  sun,  while  those  of  the  sun 
are  visible  only  to  the  small  portion  of  the  hemisphere  on  which 
the  moon's  shadow  falls,  it  happens  that  for  any  particular  place 
on  the  earth,  lunar  eclipses  are  more  frequently  visible  than  solar. 
In  any  year,  the  number  of  eclipses  of  both  luminaries  cannot  be 
ess  than  two  nor  more  than  seven  :  the  most  usual  number  is  four, 
and  it  is  very  rare  to  have  more  than  six.  A  total  eclipse  of  the 
moon  frequently  happens  at  the  next  full  moon  after  an  eclipse  of 
the  sun.  For  since,  in  an  eclipse  of  the  sun,  the  sun  is  at  or  near 
one  of  the  moon's  nodes,  the  earth's  shadow  must  be  at  or  near 
the  other  node,  and  may  not  have  passed  so  far  from  the  node  as 
the  lunar  ecliptic  limits,  before  the  moon  overtakes  it. 

270.  It  has  been  observed  already,  that  were  the  spectator  on 
the  moon  instead  of  on  the  earth,  he  would  see  the  earth  eclipsed 
by  the  moon,  and  the  calculation  of  the  eclipse  would  be  very  sim- 
ilar to  that  of  a  lunar  eclipse  ;  but  to  an  observer  on  the  earth  the 

*  Biot,  Ast.  Phys.  p.  401. 


160  THE   MOON. 

eclipse  does  not  of  course  begin  when  the  earth  first  enters  the 
moon's  shadow,  and  it  is  necessary  to  determine  not  only  what 
portion  of  the  earth's  surface  will  be  covered  by  the  moon's  sha- 
dow, but  likewise  the  path  described  by  its  center  relative  to  va- 
rious places  on  the  surface  of  the  earth.  This  is  known  when  the 
latitude  and  longitude  of  the  center  of  the  shadow  on  the  earth,  is 
determined  for  each  instant.  The  latitude  and  longitude  of  the 
moon  are  found  on  the  supposition  that  the  spectator  views  it  from 
the  center  of  the  earth,  whereas  his  position  on  the  surface  changes, 
in  consequence  of  parallax,  both  the  latitude  and  longitude,  and 
the  amount  of  these  changes  must  be  accurately  estimated,  before 
the  appearance  of  the  eclipse  at  any  particular  place  can  be  fully 
determined. 

The  details  of  the  method  of  calculating  a  solar  eclipse  cannot 
be  understood  in  any  way  so  well,  as  by  actually  performing  the 
process  according  to  a  given  example.  For  such  details  therefore 
the  reader  is  referred  to  the  Supplement. 

271.  In  total  eclipses  of  the  sun,  there  has  sometimes  been  ob- 
served a  remarkable  radiation  of  light  from  the  margin  of  the  sun. 
This  has  been  ascribed  to  an  illumination  of  the  solar  atmosphere, 
but  it  is  with  more  probability  owing  to  the  zodiacal  light  (Art. 
152,)  which  at  that  time  is  projected  around  the  sun,  and  which  is 
of  such  dimensions  as  to  extend  far  beyond  the  solar  orb.* 

A  total  eclipse  of  the  sun  is  one  of  the  most  sublime  and  impres- 
sive phenomena  of  nature.  Among  barbarous  tribes  it  is  ever  con- 
templated with  fear  and  astonishment,  while  among  cultivated  na- 
tions it  is  recognized,  from  the  exactness  with  which  the  time  of 
occurrence  and  the  various  appearances  answer  to  the  prediction, 
as  affording  one  of  the  proudest  triumphs  of  astronomy.  By 
astronomers  themselves  it  is  of  course  viewed  with  the  highest 
interest,  not  only  as  verifying  their  calculations,  but  as  contribu- 
ting to  establish  beyond  all  doubt  the  certainty  of  those  grand 
laws,  the  truth  of  which  is  involved  in  the  result.  During  the 
eclipse  of  June,  1806,  which  was  one  of  the  most  remarkable  on 

*  See  an  excellent  description  and  delineation  of  this  appearance  as  it  was  exhibited 
in  the  eclipse  of  1806,  in  the  Transactions  of  the  Albany  Institute,  by  the  late  Chan, 
eellor  De  Witt, 


LONGITUDE.  161 

record,  the  time  of  total  darkness,  as  seen  by  the  author  of  this 
work,  was  about  mid-day.  The  sky  was  entirely  cloudless,  but 
as  the  period  of  total  obscuration  approached,  a  gloom  pervaded 
all  nature.  When  the  sun  was  wholly  lost  sight  of,  planets  and 
stars  came  into  view ;  a  fearful  pall  hung  upon  the  sky,  unlike 
both  to  night  and  to  twilight ;  and,  the  temperature  of  the  air  rap- 
idly declining,  a  sudden  chill  came  over  the  earth.  Even  the  ani- 
mal tribes  exhibited  tokens  of  fear  and  agitation. 

From  1831  to  1838  was  a  period  remarkable  for  great  eclipses 
of  the  sun,  in  which  time  there  were  no  less  than  five  of  the  most 
remarkable  character.  The  next  total  eclipse  of  the  sun,  visible 
in  the  United  States,  will  occur  on  the  7th  of  August,  1869. 


CHAPTER  VIII. 

LONGITUDE TIDES. 

272.  As  eclipses  of  the  sun  afford  one  of  the  most  approved 
methods  of  finding  the  longitudes  of  places,  our  attention  is  natu- 
rally turned  next  towards  that  subject. 

The  ancients  studied  astronomy  in  order  that  they  might  read 
their  destinies  in  the  stars  :  the  moderns,  that  they  may  securely 
navigate  the  ocean.  A  large  portion  of  the  refined  labors  of 
modern  astronomy,  has  been  directed  towards  perfecting  the  as- 
tronomical tables  with  the  view  of  finding  the  longitude  at  sea, — 
an  object  manifestly  worthy  of  the  highest  efforts  of  science,  con- 
sidering the  vast  amount  of  property  and  of  human  life  involved 
in  navigation. 

273.  The  difference  of  longitude  between  two  places  may  be  found 
by  any  method,  by  which  we  can  ascertain  the  difference  of  their  local 
times,  at  the  same  instant  of  absolute  time. 

As  the  earth  turns  on  its  axis  from  west  to  east,  any  place  that 
lies  eastward  of  another  will  come  sooner  under  the  sun,  or  will 

21 


162  THE  MOON. 

have  the  sun  earlier  on  the  meridian,  and  consequently,  in  respect 
to  the  hour  of  the  day,  will  be  in  advance  of  the  other  at  the 
rate  of  one  hour  for  every  15°,  or  four  minutes  of  time  for  each 
degree.  Thus,  to  a  place  15°  east  of  Greenwich,  it  is  1  o'clock, 
P.  M.  when  it  is  noon  at  Greenwich;  and  to  a  place  15°  west  of 
that  meridian,  it  is  11  o'clock,  A.  M.  at  the  same  instant.  Hence, 
the  difference  of  time  at  any  two  places,  indicates  their  difference 
of  longitude. 

274.  The  easiest  method  of  finding  the  longitude  is  by  means 
of  an  accurate  time  piece,  or  chronometer.  Let  us  set  out  from 
London  with  a  chronometer  accurately  adjusted  to  Greenwich 
time,  and  travel  eastward  to  a  certain  place,  where  the  time  is 
accurately  kept,  or  may  be  ascertained  by  observation.  We  find, 
for  example,  that  it  is  1  o'clock  by  our  chronometer,  when  it  is 
2  o'clock  and  30  minutes  at  the  place  of  observation.  Hence, 
the  longitude  is  15x1.5=22^°  E.  Had  we  travelled  westward 
until  our  chronometer  was  an  hour  and  a  half  in  advance  of  the 
time  at  the  place  of  observation,  (that  is,  so  much  later  in  the 
day,)  our  longitude  would  have  been  22|°  W.  But  it  would  not 
be  necessary  to  repair  to  London  in  order  to  set  our  chronometer 
to  Greenwich  time.  This  might  be  done  at  any  observatory,  or 
any  place  whose  longitude  had  been  accurately  determined.  For 
example,  the  time  at  New  York  is  4h.  56m.  48.5  behind  that  of 
Greenwich.  If,  therefore,  we  set  our  chronometer  so  much  be- 
fore the  true  time  at  New  York,  it  will  indicate  the  time  at  Green- 
wich. Moreover,  on  arriving  at  different  places,  any  where  on 
the  earth,  whose  longitude  is  accurately  known,  we  may  learn 
whether  our  chronometer  keeps  accurate  time  or  not,  and  if  not, 
the  amount  of  its  error.  Chronometers  have  been  constructed  of 
such  an  astonishing  degree  of  accuracy,  as  to  deviate  but  a  few 
seconds  in  a  voyage  from  London  to  Baffin's  Bay  and  back,  during 
an  absence  of  several  years.  But  chronometers  which  are  suffi- 
ciently accurate  to  be  depended  on  for  long  voyages,  are  too  ex- 
pensive for  general  use,  and  the  means  of  verifying  their  accuracy 
are  not  sufficiently  easy.  Moreover,  chronometers  by  being  trans- 
ported from  one  place  to  another,  change  their  daily  rate,  or  de- 
part from  that  mean  rate  which  they  preserve  at  a  fixed  station. 


LONGITUDE.  1 63 

A  chronometer,  therefore,  cannot  be  relied  on  for  determining  the 
longitudes  of  places  where  the  greatest  degree  of  accuracy  is  re- 
quired, especially  where  the  instrument  is  conveyed  over  land, 
although  the  uncertainty  attendant  on  one  instrument  may  be 
nearly  obviated  by  employing  several  and  taking  their  mean 
results.* 

275.  Eclipses  of  the  sun  and  moon  are  sometimes  used  for  de- 
termining the  longitude.     The  exact  instant  of  immersion  or  of 
emersion,  or  any  other  definite  moment  of  the  eclipse  which  pre- 
sents itself  to  two  distant  observers,  affords  the  means  of  com- 
paring their  difference  of  time,  and  hence  of  determining  their 
difference  of  longitude.     Since  the  entrance  of  the  moon  into 
the  earth's  shadow,  in  a  lunar  eclipse,  is  seen  at  the  same  instant 
of  absolute  time  at  all  places  where  the  eclipse  is  visible,  (Art. 
262,)  this  observation  would  be  a  very  suitable  one  for  finding 
the  longitude  were  it  not  that,  on  account  of  the  increasing  dark- 
ness of  the  penumbra  near  the  boundaries  of  the  shadow,  it  is 
difficult  to  determine  the  precise  instant  when  the  moon  enters  the 
shadow.     By  taking  observations  on  the  immersions  of  known 
spots  on  the  lunar  disk,  a  mean  result  may  be  obtained  which  will 
give  the  longitude  with  tolerable  accuracy.     In  an  eclipse  of  the 
sun,  the  instants  of  immersion  and  emersion  may  be  observed  with 
greater  accuracy,  although,  since  these  do  not  take  place  at  the 
same  instant  of  absolute  time,  the  calculation  of  the  longitude  from 
observations  on  a  solar  eclipse  are  complicated  and  laborious. 

A  method  very  similar  to  the  foregoing,  by  observations  on 
eclipses  of  Jupiter's  satellites,  and  on  occultations  of  stars,  will 
be  mentioned  hereafter. 

276.  The  Lunar  method  of  finding  the  longitude,  at  sea,  is  in 
many  respects  preferable  to  every  other.     It  consists  in  measuring 
(with  a  sextant)  the  angular  distance  between  the  moon  and  the 
sun,  or  between  the  moon  and  a  star,  and  then  turning  to  the  Nau- 
tical Almanac,f  and  finding  what  time  it  was  at  Greenwich  when 

*  Woodhouse,  p.  838. 

t  The  Nautical  Almanac  is  a  book  published  annually  by  the  British  Board  of 
Longitude,  containing  various  tables  and  astronomical  information  for  the  use  oi 


164  THE   MOON. 

that  distance  was  the  same.  The  moon  moves  so  rapidly,  that  this 
distance  will  not  be  the  same  except  at  very  nearly  the  same  in- 
stant of  absolute  time.  For  example,  at  9  o'clock,  A.  M.,  at  a  cer- 
tain place,  we  find  the  angular  distance  of  the  moon  and  the  sun  to 
be  72° ;  and  on  looking  into  the  Nautical  Almanac,  we  find  that 
at  the  time  when  this  distance  was  the  same  for  the  meridian  of 
Greenwich  was  1  o'clock,  P.  M. ;  hence  we  infer  that  the  longi- 
tude of  the  place  is  four  hours,  or  60°  west. 

The  Nautical  Almanac  contains  the  true  angular  distance  of 
the  moon  from  the  sun,  from  the  four  large  planets,  (Venus,  Mars, 
Jupiter,  and  Saturn,)  and  from  nine  bright  fixed  stars,  for  the  be- 
ginning of  every  third  hour  of  mean  time  for  the  meridian  of 
Greenwich ;  and  the  mean  time  corresponding  to  any  intermediate 
hour,  may  be  found  by  proportional  parts.* 

277.  It  would  be  a  very  simple  operation  to  determine  the  lon- 
gitude by  Lunar  Distances,  if  the  process  as  described  in  the 
preceding  article  were  all  that  is  necessary ;  but  the  various  cir- 
cumstances of  parallax,  refraction,  and  dip  of  the  horizon,  would 
differ  more  or  less  at  the  two  places,  even  were  the  bodies  whose 
distances  were  taken  in  view  from  both,  which  is  not  necessarily 
the  case.     The  observations,  therefore,  require  to  be  reduced  to 
the  center  of  the  earth,  being  cleared  of  the  effects  of  parallax  and 
refraction.     Hence,  three  observers  are  necessary  in  order  to  take 
a  lunar  distance  in  the  most  exact  manner,  viz.  two  to  measure 
the  altitudes  of  the  two  bodies  respectively,  at  the  same  time  that 
the  third  takes  the  angular  distance  between  them.     The  altitudes 
of  the  two  luminaries  at  the  time  of  observation  must  be  known, 
in  order  to  estimate  the  effects  of  parallax  and  refraction. 

278.  Although  the  lunar  method  of  finding  the  longitude  at 
sea  has  many  advantages  over  the  other  methods  in  use,  yet  it 

navigators.  The  American  Almanac  also  contains  a  variety  of  astronomical  informa- 
tion, peculiarly  interesting  to  the  people  of  the  United  States,  in  connexion  with  a 
vast  amount  of  statistical  matter.  It  is  well  deserving  a  place  in  the  library  of  the 
student. 

*  See  Bowditch's  Navigator,  Tenth  Ed.  p.  226. 


TIDES.  165 

has  also  its  disadvantages.  One  is,  the  great  exactness  requisite 
in  observing  the  distance  of  the  moon  from  the  sun  or  star,  as  a 
small  error  in  the  distance  makes  a  considerable  error  in  the  longi- 
tude. The  moon  moves  at  the  rate  of  about  a  degree  in  two 
hours,  or  one  minute  of  space  in  two  minutes  of  time.  There- 
fore, if  we  make  an  error  of  one  minute  in  observing  the  distance, 
we  make  an  error  of  two  minutes  in  time,  or  30  miles  of  longitude 
at  the  equator.  A  single  observation  with  the  best  sextants,  may 
be  liable  to  an  error  of  more  than  half  a  minute  ;  but  the  accuracy 
of  the  result  may  be  much  increased  by  a  mean  of  several  obser- 
vations taken  to  the  east  and  west  of  the  moon.  The  imperfection 
of  lunar  tables  was  until  recently  considered  as  an  objection  to  this 
method.  Until  within  a  few  years,  the  best  lunar  tables  were 
frequently  erroneous  to  the  amount  of  one  minute,  occasioning  an 
error  of  30  miles.  The  error  of  the  best  tables  now  in  use  will 
rarely  exceed  7  or  8  seconds.* 


TIDES. 

279.  The  tides  are  an  alternate  rising  and  falling  of  the  waters 
of  the  ocean,  at  regular  intervals.  They  have  a  maximum  and  a 
minimum  twice  a  day,  twice  a  month,  and  twice  a  year.  Of  the 
daily  tide,  the  maximum  is  called  High  tide,  and  the  minimum 
Low  tide.  The  maximum  for  the  month  is  called  Spring  tide,  and 
the  minimum  Neap  tide.  The  rising  of  the  tide  is  called  Flood 
and  its  falling  Ebb  tide. 

Similar  tides,  whether  high  or  low,  occur  on  opposite  sides  of 
the  earth  at  once.  Thus  at  the  same  time  it  is  high  tide  at  any 
given  place,  it  is  also  high  tide  on  the  inferior  meridian,  and  the 
same  is  true  of  the  low  tides. 

The  interval  between  two  successive  high  tides  is  12h.  25m. ; 
or,  if  the  same  tide  be  considered  as  returning  to  the  meridian, 
after  having  gone  around  the  globe,  its  return  is  about  50  minutes 
later  than  it  occurred  on  the  preceding  day.  In  this  respect,  as 
well  as  in  various  others,  it  corresponds  very  nearly  to  the  motions 
of  the  moon. 

*  Brinkley's  Elements  of  Astronomy,  p.  241 


166  THE    MOON. 

The  average  height  for  the  whole  globe  is  about  2£  feet;  01, 
if  the  earth  were  covered  uniformly  with  a  stratum  of  water,  the 
difference  between  the  two  diameters  of  the  oval  would  be  5  feet, 
or  more  exactly  5  feet  and  8  inches  ;  but  its  natural  height  at 
various  places  is  very  various,  sometimes  rising  to  60  or  70  feet, 
and  sometimes  being  scarcely  perceptible.  At  the  same  place 
also  the  phenomena  of  the  tides  are  very  different  at  different 
times. 

Inland  lakes  and  seas,  even  those  of  the  largest  class,  as  Lake 
Superior,  or  the  Caspian,  have  no  perceptible  tide. 

280.  Tides  are  caused  by  the  unequal  attraction  of  the  sun  and 
moon  upon  different  parts  of  the  earth. 

Suppose  the  projectile  force  by  which  the  earth  is  carried  for- 
ward in  her  orbit,  to  be  suspended,  and  the  earth  to  fall  towards 
one  of  these  bodies,  the  moon,  for  example,  in  consequence  of 
their  mutual  attraction.  Then,  if  all  parts  of  the  earth  fell 
equally  towards  the  moon,  no  derangement  of  its  different  parts 
would  result,  any  more  than  of  the  particles  of  a  drop  of  water 
in  its  descent  to  the  ground.  But  if  one  part  fell  faster  than  an- 
other, the  different  portions  would  evidently  be  separated  from 
each  other.  Now  this  is  precisely  what  takes  place  with  respect 
to  the  earth  in  its  fall  towards  the  moon.  The  portions  of  the 
earth  in  the  hemisphere  next  to  the  moon,  on  account  of  being 
nearer  to  the  center  of  attraction,  fall  faster  than  those  in  the  op- 
posite hemisphere,  and  consequently  leave  them  behind.  The 
solid  earth,  on  account  of  its  cohesion,  cannot  obey  this  impulse, 
since  all  its  different  portions  constitute  one  mass,  which  is  acted 
on  in  the  same  manner  as  though  it  were  all  collected  in  the  cen- 
ter ;  but  the  waters  on  the  surface,  moving  freely  under  this  im- 
pulse, endeavor  to  desert  the  solid  mass  and  fall  towards  the 
moon.  For  a  similar  reason  the  waters  in  the  opposite  hemisphere 
falling  less  towards  the  moon  than  the  solid  earth,  are  left  behind, 
or  appear  to  rise  from  the  center  of  the  earth. 

281.  Let  DEFG  (Fig.  56,)  represent  the  globe  ;  and,  for  the  sake 
of  illustrating  the  principle,  we  will  suppose  the  waters  entirely  to 
cover  the  globe  at  a  uniform  depth.     Let  defg  represent  the  solid 


TIDES. 


167 


globe,  and  the  circular  ring  exterior  to  Fig.  56. 

it,  the  covering  of  waters.  Let  C  be 
the  center  of  gravity  of  the  solid  mass, 
A  that  of  the  hemisphere  next  to  the 
moon,  and  B  that  of  the  remoter  hemi- 
sphere. Now  the  force  of  attraction 
exerted  by  the  moon,  acts  in  the  same 
manner  as  though  the  solid  mass  were 
all  concentrated  in  C,  and  the  waters 
of  each  hemisphere  at  A  and  B  respec- 
tively ;  and  (the  moon  being  supposed  above  E)  it  is  evident  that 
A  will  tend  to  leave  C,  and  C  to  leave  B  behind.  The  same  must 
evidently  be  true  of  the  respective  portions  of  matter,  of  which 
these  points  are  the  centers  of  gravity.  The  waters  of  the  globe 
will  thus  be  reduced  to  an  oval  shape,  being  elongated  in  the  direc- 
tion of  that  meridian  which  is  under  the  moon,  and  flattened  in 
the  intermediate  parts,  and  most  of  all  at  points  ninety  degrees  dis- 
tant from  that  meridian. 

Were  it  not,  therefore,  for  impediments  which  prevent  the  force 
from  producing  its  full  effects,  we  might  expect  to  see  the  great 
tide-wave,  as  the  elevated  crest  is  called,  always  directly  beneath 
the  moon,  attending  it  regularly  around  the  globe.  But  the  in- 
ertia of  the  waters  prevents  their  instantly  obeying  the  moon's 
attraction,  and  the  friction  of  the  waters  on  the  bottom  of  the 
ocean,  still  further  retards  its  progress.  It  is  not  therefore  until 
several  hours  (differing  at  different  places)  after  the  moon  has 
passed  the  meridian  of  a  place,  that  it  is  high  tide  at  that  place. 

282.  The  sun  has  a  similar  action  to  the  moon,  but  only  one 
third  as  great.  On  account  of  the  great  mass  of  the  sun  com- 
pared with  that  of  the  moon,  we  might  suppose  that  his  action 
in  raising  the  tides  would  be  greater  than  the  moon's ;  but  the 
nearness  of  the  moon  to  the  earth  more  than  compensates  for 
the  sun's  greater  quantity  of  matter.  Let  us,  however,  form  a 
just  conception  of  the  advantage  which  the  moon  derives  from  her 
proximity.  It  is  not  that  her  actual  amount  of  attraction  is  thus 
rendered  greater  than  that  of  the  sun ;  but  it  is  that  her  attraction 
for  the  different  parts  of  the  earth  is  very  unequal,  while  that  of 


168  THE  MOON. 

the  sun  is  nearly  uniform.  It  is  the  inequality  of  this  action,  and 
not  the  absolute  force,  that  produces  the  tides.  The  diameter  of 
the  earth  is  ^  of  the  distance  of  the  moon,  while  it  is  less  than 
roioT  of  the  distance  of  the  sun. 

283.  Having  now  learned  the  general  cause  of  the  tides,  we 
will  next  attend  to  the  explanation  of  particular  phenomena. 

The  Spring  tides,  or  those  which  rise  to  an  unusual  height 
twice  a  month,  are  produced  by  the  sun  and  moon's  acting  to- 
gether; and  the  Neap  tides,  or  those  which  are  unusually  low 
twice  a  month,  are  produced  by  the  sun  and  moon's  acting  in 
opposition  to  each  other.  The  Spring  tides  occur  at  the  syzygies ; 
the  Neap  tides  at  the  quadratures.  At  the  time  of  new  moon, 
the  sun  and  moon  both  being  on  the  same  side  of  the  earth,  and 
acting  upon  it  in  the  same  line,  their  actions  conspire,  and  the 
sun  may  be  considered  as  adding  so  much  to  the  force  of  the 
moon.  We  have  already  explained  how  the  moon  contributes  to 
raise  a  tide  on  the  opposite  side  of  the  earth.  But  the  sun  as  well 
as  the  moon  raises  its  own  tide-wave,  which,  at  new  moon,  coin- 
cides with  the  lunar  tide-wave.  At  full  moon,  also,  the  two  lumina- 
ries conspire  in  the  same  way  to  raise  the  tide  ;  for  we  must  recol- 
lect that  each  body  contributes  to  raise  the  tide  6n  the  opposite 
side  of  the  earth  as  well  as  on  the  side  nearest  to  it.  At  both  the 
conjunctions  and  oppositions,  therefore,  that  is,  at  the  syzygies, 
we  have  unusually  high  tides.  But  here  also  the  maximum  effect 
is  not  at  the  moment  of  the  syzygies,  but  36  hours  afterwards. 

At  the  quadratures,  the  solar  wave  is  lowest  where  the  lunar 
wave  is  highest ;  hence  the  low  tide  produced  by  the  sun  is  sub- 
tracted from  high  water  and  produces  the  Neap  tides.  Moreover, 
at  the  quadratures  the  solar  wave  is  highest  where  the  lunar  wave 
is  lowest,  and  hence  is  to  be  added  to  the  height  of  low  water  at 
the  time  of  Neap  tides.  Hence  the  difference  between  high  and 
low  water  is  only  about  half  as  great  at  Neap  tide  as  at  Spring  tide. 

284.  The  power  of  the  moon  or  of  the  sun  to  raise  the  tide  is 
found  by  the  doctrine  of  universal  gravitation  to  be  inversely  as 
the  cube  of  the  distance*     The  variations  of  distance  in  the  sun  are 

*  La  Place,  Syst,  du  Monde,  1.  iv,  c.  x. 


TIDES. 


169 


not  great  enough  to  influence  the  tides  very  materially,  but  the 
variations  in  the  moon's  distances  have  a  striking  effect.  The 
tides  which  happen  when  the  moon  is  in  perigee,  are  considerably 
greater  than  when  she  is  in  apogee ;  and  if  she  happens  to  be  in 
perigee  at  the  time  of  the  syzygies,  the  spring  tide  is  unusually 
high.  When  this  happens  at  the  equinoxes,  the  highest  tides  of 
the  year  are  produced. 

285.  The  declinations  of  the  sun  and  moon  have  a  considerable 
influence  on  the  height  of  the  tide.  When  the  moon,  for  example, 
has  no  declination,  or  is  in  the  equator,  as  in  figure  57,*  the  rota- 
tion of  the  earth  on  its  axis  NS  will  make  the  two  tides  exactly 
equal  on  opposite  sides  of  the  earth.  Thus  a  place  which  is  car- 
ried through  the  parallel  TT'  will  have  the  height  of  one  tide  T2 
and  the  other  tide  T'3.  The  tides  are  in  this  case  greatest  at  the 
equator,  and  diminish  gradually  to  the  poles,  where  it  will  be  low 
water  during  the  whole  day.  When  the  moon  is  on  the  north  side 
of  the  equator,  as  in  figure  58,  at  her  greatest  northern  declination, 
Fig.  57.  Fig.  58. 


a  place  describing  the  parallel  TT'  will  have  T'3  for  the  height  of 
the  tide  when  the  moon  is  on  the  superior  meridian,  and  T2  for 
the  height  when  the  moon  is  on  the  inferior  meridian.  Therefore, 
all  places  north  of  the  equator  will  have  the  highest  tide  when  the 
moon  is  above  the  horizon,  and  the  lowest  when  she  is  below  it ; 
the  difference  of  the  tides  diminishing  towards  the  equator,  where 

*  Diagrams  like  these  are  apt  to  mislead  the  learner,  by  exhibiting  the  protuberance 
occasioned  by  the  tides  as  much  greater  than  the  reality.  We  must  recollect  that  it 
amounts,  at  the  highest,  to  only  a  very  few  feet  in  eight  thousand  miles.  Were  the 
diagram,  therefore,  drawn  in  just  proportions,  the  alterations  of  figure  produced  by  the 
lides  would  be  wholly  insensible. 

22 


170  THE   MOON. 

they  are  equal.  In  like  manner,  places  south  of  the  equator  have 
the  highest  tides  when  the  moon  is  below  the  horizon,  and  the 
lowest  when  she  is  above  it.  When  the  moon  is  at  her  greatest 
declination,  the  highest  tides  will  take  place  towards  the  tropics. 
The  circumstances  are  all  reversed  when  the  moon  is  south  of  the 
equator.* 

280.  The  motion  of  the  tide- wave,  it  should  be  remarked,  is  not 
a  progressive  motion,  but  a  mere  undulation,  and  is  to  be  carefully 
distinguished  from  the  currents  to  which  it  gives  rise.  If  the 
ocean  completely  covered  the  earth,  the  sun  and  moon  being  in  the 
equator,  the  tide-wave  would  travel  at  the  same  rate  as  the  earth 
on  its  axis.  Indeed,  the  correct  way  of  conceiving  of  the  tide- 
wave,  is  to  consider  the  moon  at  rest,  and  the  earth  in  its  rotation 
from  west  to  east  as  bringing  successive  portions  of  water  under 
the  moon,  which  portions  being  elevated  successively  at  the  same 
rate  as  the  earth  revolves  on  its  axis,  have  a  relative  motion  west- 
ward in  the  same  degree. 

287.  The  tides  of  rivers,  narrow  bays,  and  shores  far  from  the 
main  body  of  the  ocean,  are  not  produced  in  those  places  by  the 
direct  action  of  the  sun  and  moon,  but  are  subordinate  waves 
propagated  from  the  great  tide-wave. 

Lines  drawn  through  all  the  adjacent  parts  of  any  tract  of  wa- 
ter, which  have  high  water  at  the  same  time,  are  called  cotidal 
lines.^  We  may,  for  instance,  draw  a  line  through  all  places  in 
the  Atlantic  Ocean  which  have  high  tide  on  a  given  day  at  1  o'clock, 
and  another  through  all  places  which  have  high  tide  at  2  o'clock. 
The  cotidal  line  for  any  hour  may  be  considered  as  representing 
the  summit  or  ridge  of  the  tide- wave  at  that  time  ;  and  could  the 
spectator,  detached  from  the  earth,  perceive  the  summit  of  the 
wave,  he  would  see  it  travelling  round  the  earth  in  the  open  ocean 
once  in  twenty  four  hours,  followed  by  another  twelve  hours  dis- 
tant, and  both  sending  branches  into  rivers,  bays,  and  other  open- 
ings into  the  main  land.  These  latter  are  called  Derivative  tides, 


*  Edin.  Encyc.  Art.  Astronomy,  p.  623. 

t  Whewell,  Phil.  Transaction  for  1833,  p.  148. 


TIDES.  171 

while  those  raised  directly  by  the  action  of  the  sun  and  moon,  are 
called  Primitive  tides. 

288.  The  velocity  with  which  the  wave  moves  will  depend  on 
various  circumstances,  but  principally  on  the  depth,  and  probably 
on  the  regularity  of  the  channel.     If  the  depth  be  nearly  uniform, 
the  cotidal  lines  will  be  nearly  straight  and  parallel.     But  if  some 
parts  of  the  channel  are  deep  while  others  are  shallow,  the  tide 
will  be  detained  by  the  greater  friction  of  the  shallow  places,  and 
the  cotidal  lines  will  be  irregular.     The  direction  also  of  the  de- 
rivative tide,  may  be  totally  different  from  that  of  the  primitive. 
Thus,  (Fig.  59,)  if  the  great  tide-  Fig.  59. 

wave,  moving  from  east  to  west, 
be  represented  by  the  lines  1,  2, 
3, 4,  the  derivative  tide  which  is 
propagated  up  a  river  or  bay, 
will  be  represented  by  the  cotidal 
lines  3,  4,  5,  6,  7.  Advancing 
faster  in  the  channel  than  next 
the  banks,  the  tides  will  lag  be- 
hind towards  the  shores,  and  the 
cotidal  lines  will  take  the  form 
of  curves  as  represented  in  the 
diagram. 

289.  On  account  of  the  retarding  influence  of  shoals,  and  an 
uneven,  indented  coast,  the  tide-wave  travels  more  slowly  along 
the  shores  of  an  island  than  in  the  neighboring  sea,  assuming  con- 
vex figures  at  a  little  distance  from  the  island  and  on  opposite 
sides  of  it.     These   convex  lines  sometimes  meet  and,  become 
blended  in  such  a  manner  as  to  create  singular  anomalies  in  a  sea 
much  broken  by  islands,  as  well  as  on  coasts  indented  with  numer- 
ous tfays  arid  rivers.*     Peculiar  phenomena  are  also  produced, 
when  the  tide  flows  in  at  opposite  extremities  of  a  reef  or  island, 
as  into  the  two  opposite  ends  of  Long  Island  Sound.     In  certain 

*  See  an  excellent  representation  and  description  of  these  different  phenomena  by 
Professor  Whewell,  Phil.  Trans.  1833,  p.  153. 


172  THE    MOON. 

cases  a  tide-wave  is  forced  into  a  narrow  arm  of  the  sea,  and 
produces  very  remarkable  tides.  The  tides  of  the  Bay  of  Fundy 
(the  highest  in  the  world)  sometimes  rise  to  the  height  of  60  or  70 
feet ;  and  the  tides  of  the  river  Severn,  near  Bristol  in  England, 
rise  to  the  height  of  40  feet. 

290.  The   Unit  of  Altitude  of  any  place,  is  the  height  of  the 
maximum  tide  after  the  syzygies,  (Art.  283,)  being  usually  about 
36  hours  after  the  new  or  full  moon.    But  as  the  amount  of  this 
tide  would  be  affected  by  the  distance  of  the  sun  and  moon  from 
the  earth,  (Art.  284,)  and  by  their  declinations,  (Art.  285,)  these 
distances  are  faken  at  their  mean  value,  and  the  luminaries  are 
supposed  to  be  in  the  equator  ;  the  observations  being  so  reduced 
as  to  conform  to  these  circumstances.     The  unit  of  altitude  can  be 
ascertained  by  observation  only.     The  actual  rise  of  the  tide  de- 
pends much  on  the  strength  and  direction  of  the  wind.     When 
high  winds  conspire  with  a  high  flood  tide,  as  is  frequently  the 
case  near  the  equinoxes,  the  tide  rises  to  a  very  unusual  height. 
We  subjoin  from  the  American  Almanac  a  few  examples  of  the 
unit  of  altitude  for  different  places. 

Feet. 

Cumberland,  head  of  the  Bay  of  Fundy,  71 

Boston,  .  .  .  .  11J 

New  Haven,  ....  8 
New  York,  ....  5 
Charleston,  S.  C.,  ...  6 

291.  The  Establishment  of  any  port  is  the  mean  interval  between 
noon  and  the  time  of  high  water,  on  the  day  of  new  or  full  moon. 
As  the  interval  for  any  given  place  is  always  nearly  the  same,  it 
becomes  a  criterion  of  the  retardation  of  the  tides  at  that  place. 
On  account  of  the  importance  to  navigation  of  a  correct  know- 
ledge of  the  tides,  the  British  Board  of  Admiralty,  at  the  sugges- 
tion of  the  Royal  Society,  recently  issued  orders  to  their  agents 
in  various  important  naval  stations,  to  have  accurate  observations 
made  on  the  tides,  with  the  view  of  ascertaining  the  establishment 
and  various  other  particulars  respecting  each  station;*  and  the 

*  Lubbock,  Report  on  the  Tides,  1833. 


TIDES.  173 

government  of  the  United  States  is  prosecuting  similar  investiga- 
tions respecting  our  own  ports. 

292.  According  to  Professor  Whewell,*  the  tides  on  the  coast 
of  North  America  are  derived  from  the  great  tide-wave  of  the 
South  Atlantic,  which  runs  steadily  northward  along  the  coast  to 
the  mouth  of  the  Bay  of  Fundy,  where  it  meets  the  northern  tide 
wave  flowing  in  the  opposite  direction.     Hence  he  accounts  for 
the  high  tides  of  the  Bay  of  Fundy. 

293.  The   largest  lakes  and   inland  seas  have  no  perceptible 
tides.     This  is  asserted  by  all  writers  respecting  the  Caspian  and 
Euxine,  and  the  same  is  found  to  be  true  of  the  largest  of  the 
North  American  lakes,  Lake  Superior. f 

Although  these  several  tracts  of  water  appear  large  when  taken 
by  themselves,  yet  they  occupy  but  small  portions  of  the  surface 
of  the  globe,  as  will  appear  evident  from  the  delineation  of  them 
on  an  artificial  globe.  Now  we  must  recollect  that  the  primitive 
tides  are  produced  by  the  unequal  action  of  the  sun  and  moon 
upon  the  different  parts  of  the  earth ;  and  that  it  is  only  at  points 
whose  distance  from  each  other  bears  a  considerable  ratio  to  the 
whole  distance  of  the  sun  or  the  moon,  that  the  inequality  of  ac- 
tion becomes  manifest.  The  space  required  is  larger  than  either 
of  these  tracts  of  water.  It  is  obvious  also  that  they  have  no  op- 
portunity to  be  subject  to  a  derivative  tide. 

294.  To  apply  the  theory  of  universal  gravitation  to  all  the  va- 
rying circumstances  that  influence  the  tides,  becomes  a  matter  of 
such  intricacy,  that  La  Place  pronounces  "  the  problem  of  the 
tides"  the  most  difficult  problem  of  celestial  mechanics. 

295.  The  Atmosphere  that  envelops  the  earth,  must  evidently  be 
subject  to  tfre  action  of  the  same  forces  as  the  covering  of  waters, 
and  hence  we  might  expect  a  rise  and  fall  of  the  barometer,  indi- 
cating an  atmospheric  tide  corresponding  to  the  tide  of  the  ocean. 


*  Phil.  Trans.  1833,  p.  172. 

t  See  Experiments  of  Gov.  Cass,  Am.  Jour.  Science. 


174  THE  PLANETS. 

La  Place  has  calculated  the  amount  of  this  aerial  tide.  It  is  too 
inconsiderable  to  be  detected  by  changes  in  the  barometer,  unless 
by  the  most  refined  observations.  Hence  it  is  concluded,  that  the 
fluctuations  produced  by  this  cause  are  too  slight  to  affect  me- 
teorological phenomena  in  any  appreciable  degree.* 


CHAPTER    IX. 

OF  THE  PLANETS THE  INFERIOR  PLANETS,  MERCURY  AND  VENUS. 

296.  THE  name  planet  signifies  a  wanderer^  and  is  applied  to 
this  class  of  bodies  because  they  shift  their  positions  in  the  heav 
ens,  whereas  the  fixed  stars  constantly  maintain  the  same  places 
with  respect  to  each  other.  The  planets  known  from  a  high  an- 
tiquity, are  Mercury,  Venus,  Earth,  Mars,  Jupiter,  and  Saturn. 
To  these,  in  1781,  was  added  Uranus, J  (or  Herschel,  as  it  is  some- 
times called  from  the  name  of  its  discoverer,)  and,  as  late  as  the 
commencement  of  the  present  century,  four  more  were  added, 
namely,  Ceres,  Pallas,  Juno,  and  Vesta.  These  bodies  are  desig- 
nated by  the  following  characters : 

1.  Mercury  S  7.  Ceres  ? 

2.  Venus  9  8.  Pallas  $ 

3.  Earth  0  9.  Jupiter  U 

4.  Mars  <?  10.  Saturn  ^ 

5.  Vesta  fi  11.  Uranus  ^ 

6.  Juno  $ 

The  foregoing  are  called  the  primary  planets.  Several  of  these 
have  one  or  more  attendants,  or  satellites,  which  revolve  around 
them,  as  they  revolve  around  the  sun.  The  earth  has  one  satel- 
lite, namely,  the  moon ;  Jupiter  has  four  ;  Saturn,  seven  ;  and  Ura- 

*  Bowditch's  La  Place,  II.  797. 
t  From  the  Greek, 
t  From 


DISTANCES   FROM  THE  SUN.  175 

nus,  six.  These  bodies  also  are  planets,  but  in  distinction  from  the 
others  they  are  called  secondary  planets.  Hence,  the  whole  num 
ber  of  planets  are  29,  viz.  11  primary,  and  18  secondary  planets.* 

297.  With  the  exception  of  the  four  new  planets,  these  bodies 
have  their  orbits  very  nearly  in  the  same  plane,  and  are  never  seen 
far  from  the  ecliptic.     Mercury,  whose  orbit  is  most  inclined  of 
all,  never  departs  further  from  the  ecliptic  than  about  7°,  while 
most  of  the  other  planets  pursue  very  nearly  the  same  path  with 
the  earth,  in  their  annual  revolution  around  the  sun.     The  new 
planets,  however,  make  wider  excursions  from  the  plane  of  the 
ecliptic,  amounting,  in  the  case  of  Pallas,  to  34£°. 

298.  Mercury  and  Yenus  are  called  inferior  planets,  because 
they  have  their  orbits  nearer  to  the  sun  than  that  of  the  earth ; 
while  all  the  others,  being  more  distant  from  the  sun  than  the 
earth,  are  called  superior  planets.     The  planets  present  great  di- 
versities among  themselves  in  respect  to  distance  from  the  sun, 
magnitude,  time  of  revolution,  and  density.     They  differ  also  in 
regard  to  satellites,  of  which,  as  we  have  seen,  three  have  respec- 
tively four,  six,  and  seven,  while  more  than  half  have  none  at  all. 
It  will  aid  the  memory,  and  render  our  view  of  the  planetary  sys- 
tem more  clear  and  comprehensive,  if  we  classify,  as  far  as  possi- 
ble, the  various  particulars  comprehended  under  the  foregoing 
heads. 

299.    DISTANCES  FROM  THE  SUN.f 


1.  Mercury, 

37,000,000 

0.3870981 

2.  Venus, 

68,000,000 

0.7233316 

3.  Earth, 

95,000,000 

1.0000000 

4.  Mars, 

142,000,000 

*1.5236923 

5.  Vesta, 

225,000,000 

2.3678700 

6.  Juno, 

) 

2.6690090 

7.  Ceres, 

>  261,000,000 

2.7672450 

8.  Pallas, 

) 

2.7728860 

*  See  Article  V.  of  the  Addenda. 

t  The  distance  in  miles,  as  expressed  in  the  first  column,  in  round  numbers,  is  to  be 
treasured  up  in  the  memory,  while  the  second  column  expresses  the  relative  distances, 
that  of  the  earth  being  1,  from  which  a  more  exact  determination  may  be  made,  when 
required,  the  earth's  distance  being  taken  at  94,885,491.  (Daily.) 


176  THE  PLANETS. 

9.  Jupiter,  485,000,000  5.2027760 

10.  Saturn,  890,000,000  9.5387861 

11.  Uranus,          1800,000,000  19.1823900 

The  dimensions  of  the  planetary  system  are  seen  from  this 
table  to  be  vast,  comprehending  a  circular  space  thirty-six  hun- 
dred millions  of  miles  in  diameter.  A  railway  car,  travelling  con- 
stantly at  the  rate  of  20  miles  an  hour,  would  require  more  than 
20,000  years  to  cross  the  orbit  of  Uranus. 

It  may  aid  the  memory  to  remark,  that  in  regard  to  the  planets 
nearest  the  sun,  the  distances  increase  in  an  arithmetical  ratio, 
while  those  most  remote  increase  in  a  geometrical  ratio.  Thus, 
if  we  add  30  to  the  distance  of  Mercury,  it  gives  us  nearly  that  of 
Venus  ;  30  more  gives  that  of  the  Earth  ;  while  Saturn  is  nearly 
twice  the  distance  of  Jupiter,  and  Uranus  twice  the  distance  of 
Saturn.  Between  the  orbits  of  Mars  and  Jupiter,  a  great  chasm 
appeared,  which  broke  the  continuity  of  the  series ;  but  the  dis- 
covery of  the  new  planets  has  filled  the  void.  A  more  exact  law 
of  the  series  was  discovered  a  few  years  since  by  Mr.  Bode  of 
Berlin.  It  is  as  follows  :  if  we  represent  the  distance  of  Mercury 
by  4,  and  increase  each  term  by  the  product  of  3  into  a  certain 
power  of  2,  we  shall  obtain  the  distances  of  each  of  the  planets  in 
succession.  Thus, 

Mercury,  ...»  4  =4 

Venus,  ....  4+3.2°  =     7 

Earth,  ....  4+3.21  =  10 

Mars,  ....  4+S.22  =  16 

Ceres,  ....  4+3.23  =  28 

Jupiter,  .         .         .         .  ^  4+3.24  =  52 

Saturn,  .         .         .        .  *  4+3.25  =100 

Uranus,  ....  4+3.26  =196 

For  example,  by  this  law,  the  distances  of  the  Earth  and  Jupi- 
ter are  to  each  other  as  10  to  52.  Their  actual  distances  as  given 
in  the  table  (Art.  299,)  are  as  1  to  5.202776 ;  but  1  :  5.202776  :  : 
10  :  52  nearly. 

The  mean  distances  of  the  planets  from  the  sun,  may  also  be  de- 
termined by  means  of  Kepler's  law,  that  the  squares  of  the  period- 


MAGNITUDES.  177 

ical  times  are  as  the  cubes  of  the  distances,  (Art.  192.)  Thus  the 
earth's  distance  being  previously  ascertained  by  means  of  the 
sun's  horizontal  parallax,  (Art.  87,)  and  the  period  of  any  other 
planet  as  Jupiter,  being  learned  from  observation,  we  say  as 
365T2562  :  4332.585**  :  :  I3 :  5.2023.  But  5.202  is  the  number, 
which,  according  to  the  table,  (Art.  299,)  expresses  the  distance  of 
Jupiter  from  the  sun. 

300.    MAGNITUDES. 

Diam.  in  Miles.       Mean  apparent  Diam.    Volume. 

Mercury,         .         .         .  3140  6".9  ^ 

Venus,    ....  7700  16".9  TV 

Earth,     ....  7912  1 

Mars,      ....  4200  6".3  | 

Ceres,     ....  160  0".5 

Jupiter,  ....  89000  36".7  1281 

Saturn    ....  79000  16".2  995 

Uranus  ....  35000  4".0  80 

We  remark  here  a  great  diversity  in  regard  to  magnitude,  a 
diversity  which  does  not  appear  to  be  subject  to  any  definite 
law.  While  Venus,  an  inferior  planet,  is  T9y  as  large  as  the  earth, 
Mars,  a  superior  planet,  is  only  |,  while  Jupiter  is  1281  times  as 
large.  Although  several  of  the  planets,  when  nearest  to  us,  appear 
brilliant  and  large  when  compared  with  the  fixed  stars,  yet  the 
angle  which  they  subtend  is  very  small,  that  of  Venus,  the  great- 
est of  all,  never  exceeding  about  1',  or  more  exactly  61".2,  and 
that  of  Jupiter  being  when  greatest  only  about  f  of  a  minute. 

The  distance  of  one  of  the  near  planets,  as  Venus  or  Mars,  may 
be  determined  from  its  parallax ;  and  the  distance  being  known, 
its  real  diameter  can  be  estimated  from  its  apparent  diameter,  in 
the  same  manner  as  we  estimate  the  diameter  of  the  sun.  (Art. 
145.) 
• __^______________ 

*  This  is  the  number  of  days  in  one  revolution  of  Jupiter. 
23 


178  THE   PLANETS. 

301.    PERIODIC  TIMES. 

Revolution  in  its  orbit  Mean  daily  motion. 

Mercury         3  months,  or          88  days,        4°    5'  32".6 
Venus,  7i      "         "          224  "  1°  36'    7//.8 

Earth,  1  year,        "          365  "  0°  59'    8".3 

Mars,  2     "  "          687  "  0°  31'  26".7 

Ceres,  4     "  "        1681  "  0°  12'  50".9 

Jupiter,        12     "  "        4332  "  0°    4'  59".3 

Saturn,        29     "  "      10759  "  0°    2'    0".6 

Uranus,        84     "  "      30686  "  0°    0'  42".4 

From  this  view,  it  appears  that  the  planets  nearest  the  sun  move 
most  rapidly.  Thus  Mercury  performs  nearly  350  revolutions 
while  Uranus  performs  one.  This  is  evidently  not  owing  merely 
to  the  greater  dimensions  of  the  orbit  of  Uranus,  for  the  length  of 
its  orbit  is  not  50  times  that  of  the  orbit  of  Mercury,  while  the 
time  employed  in  describing  it  is  350  times  that  of  Mercury.  In- 
deed this  ought  to  follow  from  Kepler's  law  that  the  squares  of 
the  periodical  times  are  as  the  cubes  of  the  distances,  from  which 
it  is  manifest  that  the  times  of  revolution  increase  faster  than  the 
dimensions  of  the  orbit.  Accordingly,  the  apparent  progress  of 
the  most  distant  planets  is  exceedingly  slow,  the  daily  rate  of  Ura- 
nus being  only  42".4  per  day ;  so  that  for  weeks  and  months,  and 
even  years,  this  planet  but  slightly  changes  its  place  among  the 
stars. 

THE  INFERIOR  PLANETS MERCURY   AND  VENUS. 

302.  The  inferior  planets,  Mercury  and  Venus,  having  their  or- 
bits so  far  within  that  of  the  earth,  appear  to  us  as  attendants  upon 
the  sun.  Mercury  never  appears  further  from  the  sun  than  29° 
(28°  48')  and  seldom  so  far ;  and  Venus  never  more  than  about 
47°  (47°  12').  Both  planets,  therefore,  appear  either  in  the  west 
soon  after  sunset,  or  in  the  east  a  little  before  sunrise.  In  high 
latitudes,  where  the  twilight  is  prolonged,  Mercury  can  seldom  be 
seen  with  the  naked  eye,  and  then  only  at  the  periods  of  its  great- 
est elongation.*  The  reason  of  this  will  readily  appear  from  the 
following  diagram. 

*  Copernicus  is  said  to  have  lamented  on  his  death-bed  that  he  had  never  been  able 
to  obtain  a  sight  of  Mercury,  and  Delambre  saw  it  but  twice. 


INFERIOR  PLANETS MERCURY  AND  VENUS. 


179 


Let  S  (Fig.  60,)  represent  the  sun,  ADB  the  orbit  of  Mercury, 
and  E  the  place  of  the  Earth.  Each  of  the  planets  is  seen  at  its 
greatest  elongation,  when  a  line,  EA  or  EB  in  the  figure,  is  a  tan- 
gent to  its  orbit.  Then  the  sun  being  at  S'  in  the  heavens,  the 
planet  will  be  seen  at  A'  and  B',  when  at  its  greatest  elongations, 
and  will  appear  no  further  from  the  sun  than  the  arc  S'A'  or  S'B' 
respectively. 

Fig.  60. 


303.  A  planet  is  said  to  be  in  conjunction  with  the  sun,  when  it 
is  seen  in  the  same  part  of  the  heavens  with  the  sun,  or  when  it 
has  the  same  longitude.     Mercury  and  Venus  have  each  two  con- 
junctions, the  inferior  and  the  superior.     The  inferior  conjunction 
is  its  position  when  in  conjunction  on  the  same  side  of  the  sun 
with  the  earth,  as  at  C  in  the  figure  :  the  superior  conjunction  is  its 
position  when  on  the  side  of  the  sun  most  distant  from  the  earth, 
as  at  D. 

304.  The  period  occupied  by  a  planet  between  two  successive 
conjunctions   with  the  earth,  is  called  its  si/nodical  revolution. 
Both  the  planet  and  the  earth  being  in  motion,  the  time  of  the 
synodical  revolution  exceeds  that  of  the  sidereal  revolution  of 
Mercury  or  Venus ;  for  when  the  planet  comes  round  to  the  place 
where  it  before  overtook  the  earth,  it  does  not  find  the  earth  at 
that  point,  byt  far  in  advance  of  it.     Thus,  let  Mercury  come  into 


180  THE   PLANETS. 

inferior  conjunction  with  the  earth  at  C,  (Fig.  60.)  In  about  88 
days,  the  planet  will  come  round  to  the  same  point  again;  but 
meanwhile  the  earth  has  moved  forward  through  the  arc  EE',  and 
will  continue  to  move  while  the  planet  is  moving  more  rapidly  to 
overtake  her,  the  case  being  analogous  to  that  of  the  hour  and 
second  hand  of  a  clock. 

Having  the  sidereal  period  of  a  planet,  (which  may  always  be 
accurately  determined  by  observation,)  we  may  ascertain  its  sy- 
nodical  period  as  follows.  Let  T  denote  the  sidereal  period  of 
the  earth,  and  T7  that  of  the  planet,  Since,  in  the  time  T  the 

T' 

earth  describes  a  complete  revolution,  T  :  T' : :  1  :  -^  =  the  part 

of  the  circumference  described  by  the  earth  in  the  time  Tf.  But 
during  the  same  time  the  planet  describes  a  whole  circumference. 

T' 

Therefore,  1  —-^  is  what  the  planet  gains  on  the  earth  in  one 

revolution.  In  order  to  a  new  conjunction  the  planet  must  gain 
an  entire  circumference  ;  therefore,  denoting  the  synodical  period 
by  S,  the  gain  in  one  revolution  will  be  to  the  time  in  which  it 
is  acquired,  as  a  whole  circumference  is  to  the  time  in  which  that 
is  gained,  which  is  the  synodical  period.  That  is, 

TV  rrrr/ 

1  •  T'  ••  1  -S  — 

*         ryi    •    •*•     »•**,•          'T' 'TV* 

From  this  formula  we  may  find  the  synodical  revolution  of  Mer- 
cury or  Venus,  by  substituting  the  numbers  denoted  by  the  letters. 

Thus,  365'256X87'969=:115.877,  which  is  the  synodical  period 

277.287 

of  Mercury. 

By  a  similar  computation,  the  synodical  revolution  of  Venus 
will  be  found  to  be  about  584  days. 

305.  The  motion  of  an  inferior  planet  is  direct  in  passing  through 
its  superior  conjunction,  and  retrograde  in  passing  through  its  infe- 
rior conjunction.  Thus  Venus,  while  going  from  B  through  D  to 
A,  (Fig.  60,)  moves  in  the  order  of  the  signs,  or  from  west  to  east, 
and  would  appear  to  traverse  the  celestial  vault  B'S'A'  from  right 
to  left ;  but  in  passing  from  A  through  C  to  B,  her  course  would 
be  retrograde,  returning  on  the  same  arc  from  left  to  right.  If 


INFERIOR    PLANETS MERCURY    AND    VENUS.  181 

the  earth  were  at  rest,  therefore,  (and  the  sun,  of  course,  at  rest,) 
the  inferior  planets  would  appear  to  oscillate  backwards  and  for- 
wards across  the  sun.  But,  it  must  be  recollected,  that  the  earth 
is  moving  in  the  same  direction  with  the  planet,  as  respects  the 
signs,  but  with  a  slower  motion.  This  modifies  the  motions  of  the 
planet,  accelerating  it  in  the  superior  and  retarding  it  in  the  infe- 
rior conjunctions.  Thus  in  figure  60,  Venus  while  moving  through 
BDA  would  seem  to  move  in  the  heavens  from  B'  to  A'  were  the 
earth  at  rest ;  but  meanwhile  the  earth  changes  its  position  from 
E  to  E',  by  which  means  the  planet  is  not  seen  at  A'  but  at  A", 
being  accelerated  by  the  arc  A' A"  in  consequence  of  the  earth's 
motion.  On  the  other  hand,  when  the  planet  is  passing  through 
its  inferior  conjunction  ACB,  it  appears  to  move  backwards  in  the 
heavens  from  A'  to  B'  if  the  earth  is  at  rest,  but  from  A'  to  B" 
if  the  earth  has  in  the  mean  time  moved  from  E  to  E',  being  re- 
tarded by  the  arc  B'B".  Although  the  motions  of  the  earth  have 
the  effect  to  accelerate  the  planet  in  the  superior  conjunction,  and 
to  retard  it  in  the  inferior,  yet,  on  account  of  the  greater  distance, 
the  apparent  motion  of  the  planet  is  much  slower  in  the  superior 
than  in  the  inferior  conjunction. 

306.  When  passing  from  the  superior  to  the  inferior  conjunction, 
or  from  the  inferior  to  the  superior  conjunction,  through  the  greatest 
elongations,  the  inferior  planets  are  stationary. 

If  the  earth  were  at  rest,  the  stationary  points  would  be  at  the 
greatest  elongations  as  at  A  and  B,  for  then  the  planet  would  be 
moving  directly  towards  or  from  the  earth,  and  would  be  seen  for 
some  time  in  the  same  place  in  the  heavens ;  but  the  earth  itself 
is  moving  nearly  at  right  angles  to  the  line  of  the  planet's  motion, 
that  is,  the  line  which  is  drawn  from  the  earth  to  the  planet  through 
the  point  of  greatest  elongation  ;  hence  a  direct  motion  is  given 
to  the  planet  by  this  cause.  When  the  planet,  however,  has  passed 
this  line,  by  its  superior  velocity  it  soon  overcomes  this  tendency 
of  the  earth  to  give  it  a  relative  motion  eastward,  and  becomes 
retrograde  as  it  approaches  the  inferior  conjunction.  Its  stationary 
point  obviously  lies  between  its  place  of  greatest  elongation,  and 
the  place  where  its  motion  becomes  retrograde.  Mercury  is  sta- 


182  THE    PLANETS. 

tionary  at  an  elongation  of  from  15°  to  20°  from  the  sun;  and 
Venus  at  about  29°.* 

307.  Mercury  and  Venus  exhibit  to  the  telescope  phases  similar  to 
those  of  the  moon. 

When  on  the  side  of  their  inferior  conjunctions,  these  planets 
appear  horned,  like  the  moon  in  her  first  and  last  quarters ;  and 
when  on  the  side  of  their  superior  conjunctions,  they  appear  gib- 
bous. At  the  moment  of  superior  conjunction,  the  whole  enlight- 
ened orb  of  the  planet  is  turned  towards  the  earth,  and  the  appear- 
ance would  be  that  of  the  full  moon,  but  the  planet  is  too  near  the 
sun  to  be  commonly  visible. 

These  different  phases  show  that  these  bodies  are  opake,  and 
shine  only  as  they  reflect  to  us  the  light  of  the  sun ;  and  the  same 
remark  applies  to  all  the  planets. 

308.  The  distance  of  an  inferior  planet  from  the  sun,  may  be 
found  by  observations  at  the  time  of  its  greatest  elongation. 

Thus  if  E  be  the  place  of  the  earth,  and  B  that  of  Venus  at  the 
time  of  her  greatest  elongation,  the  angle  SEE  will  be  known, 
being  a  right  angle.  Also  the  angle  SEE  is  known  from  observa- 
tion. Hence  the  ratio  of  SB  to  SE  becomes  known ;  or,  since  SE 
is  given,  being  the  distance  of  the  earth  from  the  sun,  SB  the  radius 
of  the  orbit  of  the  planet  is  determined.  If  the  orbits  were  both 
circles,  this  method  would  be  very  exact ;  but  being  elliptical,  we 
obtain  the  mean  value  of  the  radius  SB  by  observing  its  greatest 
elongation  in  different  parts  of  its  orbit,  f 

309.  The  orbit  of  Mercury  is  the  most  eccentric,  and  the  most 
inclined  of  all  the  planets ; J  while  that  of  Venus  varies  but  little 
from  a  circle,  and  lies  much  nearer  to  the  ecliptic. 

The  eccentricity  of  the  orbit  of  Mercury  is  nearly  }  its  semi- 
major  axis,  while  that  of  Venus  is  only  yjj  ;  the  inclination  of 
Mercury's  orbit  is  7°,  while  that  of  Venus  is  less  than  3°|.§  Mer- 
cury, on  account  of  his  different  distances  from  the  earth,  varies 


*  Herschel,  p.  242.— -Woodhouse,  557.  t  Herschel,  p.  239. 

t  The  new  planets  are  of  course  excepted.  §  Baily's  Tables. 


INFERIOR   PLANETS — MERCURY  AND    VENUS.  183 

much  in  his  apparent  diameter,  which  is  only  5"  in  the  apogee, 
but  12"  in  the  perigee. 

310.  The  most  favorable  time  for  determining  the  sidereal  revo- 
lution of  a  planet,  is  when  its  conjunction  takes  place  at  one  of 
its  nodes ;  for  then  the  sun,  the  earth,  and  the  planet,  being  in  the 
same  straight  line,  it  is  referred  to  its  true  place  in  the  heavens, 
whereas,  in  other  positions,  its  apparent  place  is  more  or  less 
affected  by  perspective. 

311.  An  inferior  planet  is  brightest  at  a  certain  point  between 
its  greatest  elongation  and  inferior  conjunction. 

Its  maximum  brilliancy  would  happen  at  the  inferior  conjunc- 
tion, (being  then  nearest  to  us,)  if  it  shined  by  its  own  light ; 
but  in  that  position,  its  dark  side  is  turned  towards  us.  Still,  its 
maximum  cannot  be  when  most  of  the  illuminated  side  is  towards 
us  ;  for  then,  being  at  the  superior  conjunction,  it  is  at  its  greatest 
distance  from  us.  The  maximum  must  therefore  be  somewhere 
between  the  two.  Venus  gives  her  greatest  light  when  about  40° 
from  the  sun. 

312.  Mercury  and  Venus  both  revolve  on  their  axes,  in  nearly  the 
same  time  with  the  earth. 

The  diurnal  period  of  Mercury  is  24h.  5m.  28s.,  and  that  of 
Venus  23h.  21m.  7s.  The  revolutions  on  their  axes  have  been 
determined  by  means  of  some  spot  or  mark  seen  by  the  telescope, 
as  the  revolution  of  the  sun  on  his  axis  is  ascertained  by  means  of 
his  spots. 

313.  Venus  is  regarded  as  the  most  beautiful  of  the  planets,  and 
is  well  known  as  the  morning  and  evening  star.     The  most  ancient 
nations  did  not  indeed  recognize  the  evening  and  morning  star  as 
one  and  the  same  body,  but  supposed  they  were  different  planets, 
and  accordingly  gave  them  different  names,  calling  the  morning 
star  Lucifer,  and  the  evening  star  Hesperus.     At  her  period  of 
greatest  splendor,  Venus  casts  a  shadow,  and  is  sometimes  visible 
in  broad  daylight.     Her  light  is  then  estimated  as  equal  to  that  of 


184  THE    PLANETS. 

twenty  stars  of  the  first  magnitude.*  At  her  period  of  greatest 
elongation,  Venus  is  visible  from  three  to  four  hours  after  the  set- 
ting or  before  rising  of  the  sun. 

314.  Every  eight  years,  Venus  forms  lier  conjunctions  with  the 
sun  in  the  same  part  of  the  heavens. 

For,  since  the  synodical  period  of  Venus  is  584  days,  and  her 
sidereal  period  224.7, 

224.7  :  360°::  584  :  935.6=the  arc  of  longitude  described  by 
Venus  between  the»first  and  second  conjunctions.  Deducting  720°, 
or  two  entire  circumferences,  the  remainder,  215°.6,  shows  how 
far  the  place  of  the  second  conjunction  is  in  advance  of  the  first. 
Hence,  in  five  synodical  revolutions,  or  2920  days,  the  same  point 
must  have  advanced  215°.6x5=1078°,  which  is  nearly  three 
entire  circumferences,  so  that  at  the  end  of  five  synodical  revolu- 
tions, occupying  2920  days,  or  8  years,  the  conjunction  of  Venus 
takes  place  nearly  in  the  same  place  in  the  heavens  as  at  first. 

Whatever  appearances  of  this  planet,  therefore,  arise  from  its 
positions  with  respect  to  the  earth  and  the  sun,  they  are  repeated 
every  eight  years  in  nearly  the  same  form. 

TRANSITS    OF    THE    INFERIOR    PLANETS. 

315.  The  Transit  of  Mercury  or  Venus,  is  its  passage  across  the 
sun's  disk,  as  the  moon  passes  over  it  in  a  solar  eclipse. 

As  a  transit  takes  place  only  when  the  planet  is  in  inferior  con- 
junction, at  which  time  her  motion  is  retrograde  (Art.  305,)  it  is 
always  from  left  to  right,  and  the  planet  is  seen  projected  on  the 
solar  disk  in  a  black  round  spot.  Were  the  orbits  of  the  inferior 
planets  coincident  with  the  plane  of  the  earth's  orbit,  a  transit 
would  occur  to  some  part  of  the  earth  at  every  inferior  conjunc- 
tion. But  the  orbit  of  Venus  makes  an  angle  of  3£°  with  the 
ecliptic,  and  Mercury  an  angle  of  7°  ;  and,  moreover,  the  apparent 
diameter  of  each  of  these  bodies  is  very  small,  both  of  which  cir- 
cumstances conspire  to  render  a  transit  a  comparatively  rare 
occurrence,  since  it  can  happen  only  when  the  sun,  at  the  time  of 

*  Francoeur,  Uranography,  p.  125. 


TRANSITS    OP    THE    INFERIOR    PLANETS.  185 

an  inferior  conjunction,  chances  to  be  at  or  extremely  near  the 
planet's  node.  The  nodes  of  Mercury  lie  in  longitude  46°  and 
226°,  points  which  the  sun  passes  through  in  May  and  November. 
It  is  only  in  these  months,  therefore,  that  transits  of  Mercury  can 
occur.  For  a  similar  reason,  those  of  Venus  occur  only  in  June 
and  December.  Since,  however,  the  nodes  of  both  planets  have 
a  small  retrograde  motion,  the  months  in  which  transits  occur  will 
change  in  the  course  of  ages. 

316.  The  intervals  between  successive  transits,  may  be  found  in 
the  following  manner.  The  formula  which  gives  the  synodical 

TxT' 
period  (Art.  304,)  is  8=^—7=,  where  S  denotes  the  period,  T  the 

sidereal  revolution  of  the  earth,  and  T7  that  of  the  planet.  If  we 
now  represent  by  m  the  number  of  synodical  periods  of  the  sun* 
in  the  required  interval,  and  by  n  the  number  of  synodical  periods 
of  the  planet ;  then,  since  the  number  of  periods  in  each  case  is  in- 
versely as  the  time  of  one,  we  have,  T  :  - — =^: :  n  :  m  .*.— — T__riv 
In  the  case  of  Mercury,  whose  sidereal  period  is  87.969  days,  while 

ftTQRQ 

that  of  the  earth  is,  365.256  days,  —  =~^^^  »  tnat   *s>  a^ter  tne 

earth  has  revolved  87969  times,  (or  after  this  number  of  years,) 
Mercury  will  have  revolved  just  277287,  and  the  two  bodies  will 
be  together  again  at  the  place  where  they  started.  But  as  periods 
of  such  enormous  length  do  not  fall  within  the.  observation  of 
man,  let  us  search  for  smaller  numbers  having  nearly  the  same 
ratio.  Now, 

87969  :  365256 : :  1  :  4£  (nearly.) 

This  shows  that  in  one  year  Mercury  will  have  made  4  revo- 
lutions and  £  of  another ;  so  that,  when  the  sun  returns  to  the  same 
node,  Mercury  will  be  more  than  60°  in  advance  of  it ;  conse- 
quently, no  transit  can  take  place  after  an  interval  of  one  year. 
But,  by  making  trial  of  2,  3,  4,  &c.  years,  we  shall  find  a  nearer 
approximation  at  the  end  of  6  years  ;  for, 

*  That  is,  the  time  in  which  the  sun  returns  again  to  the  planet's  node,  which  is  ob- 
viously after  one  year. 

24 


186  THE    PLANETS. 

87969  :  365265 ::  6  :  25— T1T.  In  6. years,  therefore,  Mercury 
will  fall  short  of  reaching  the  node  by  only  TV  of  a  revolution,  or 
about  33°.  In  13  years  the  chance  of  meeting  will  be  much 
greater,  for  in  this  period  the  earth  will  have  made  1  3  and  Mer- 
cury 54  revolutions.  The  numbers  33  and  137,  46  and  191,  afford 
a  still  nearer  approximation.* 

317.  In  a  similar  manner,  transits  of  Venus  are  probable  after 
8,  227,  235,  and  243  years.     Since  Venus  returns  to  her  conjunc- 
tion at  nearly  the  same  point  of  her  orbit,  after  8  years,  (Art.  314,) 
it  frequently  happens  that  a  transit  takes  place  after  an  interval 
of  8  years.     But  at  that  time  Venus  is  so  far  from  her  node,  that 
her  latitude  amounts  to  from  20'  to  24'.     Still  she  may  possibly 
come  within  the  sun's  disk  as  she  passes  by  him ;  for  suppose  at 
the  preceding  transit  her  latitude  was  10'  on  one  side  of  the  node 
and  is  now  10'  on  the  other  side,  this  being  less  than  the  sun's 
semi-diameter,  a  transit  may  occur  8  years  after  another.     Thus 
transits  of  Venus  took  place  in  1761  and  1769.     But  in  16  years 
the  latitude  changes  from  40'  to  48',  and  Venus  could  not  reach 
any  part  of  the  solar  disk  in  her  inferior  conjunction. 

From  the  above  series  we  should  infer  that  another  transit 
could  not  take  place  under  227  years ;  but  since  there  are  two 
nodes,  the  chance  is  doubled,  so  that  a  transit  may  occur  at  the 
other  node  in  half  that  interval,  or  in  about  113  years.  If,  at  the 
occurrence  of  the  first  transit,  Venus  had  passed  her  node,  the 
next  transit  at  the  other  node  will  happen  8  years  before  the  113 
are  completed  ;  or  if  she  had  not  reached  the  node,  it  will  happen 
8  years  later.  Hence,  after  two  transits  have  occurred  within  8 
years,  another  cannot  be  expected  before  105,  113,  or  121  years. 
Thus,  the  next  transit  will  happen  in  1874=1769+105;  also  in 
1882=1874+8. 

318.  The  great  interest  attached  by  astronomers  to  a  transit  of 
Venus,  arises  from  its  furnishing  the  most  accurate  means  in  our 
power  of  determining  the  sun's  horizontal  parallax, — an  element 
of  great  importance,  since  it  leads  us  to  a  knowledge  of  the  distance 

*  This  series  may  readily  be  obtained  by  the  method  of  Continued  Fractions.  See 
Davies's  Bourdon's  Algebra. 


TRANSITS    OF    THE    INFERIOR    PLANETS.  187 

of  the  earth  from  the  sun,  and  consequently,  by  the  application 
of  Kepler's  law,  (Art.  183,)  of  the  distances  of  all  the  other  planets. 
Hence,  in  1769,  great  efforts  were  made  throughout  the  civilized 
world,  under  the  patronage  of  different  governments,  to  observe 
this  phenomenon  under  circumstances  the  most  favorable  for  de- 
termining the  parallax  of  the  sun. 

The  method  of  finding  the  parallax  of  a  heavenly  body  described 
in  article  85,  cannot  be  relied  on  to  a  greater  degree  of  accuracy 
than  4".  In  the  case  of  the  moon,  whose  greatest  parallax  amounts 
to  about  1°,  this  deviation  from  absolute  accuracy  is  not  material ; 
but  it  amounts  to  nearly  half  the  entire  parallax  of  the  sun. 

319.  If  the  sun  and  Venus  were  equally  distant  from  us,  they 
would  be  equally  affected  by  parallax  as  viewed  by  spectators  in 
different  parts  of  the  earth,  and  hence  their  relative  situation  would 
not  be  altered  by  it ;  but  since  Venus,  at  the  inferior  conjunction, 
is  only  about  one  third  as  far  off  as  the  sun,  her  parallax  is  propor- 
tionally greater,  and  therefore  spectators  at  distant  points  will  see 
Venus  projected  on  different  parts  of  the  solar  disk,  and  as  the 
planet  traverses  the  disk,  she  will  appear  to  describe  chords  of  dif- 
ferent lengths,  by  means  of  which  the  duration  of  the  transit  may 
be  estimated  at  different  places.  The  difference  in  the  duration 
of  the  transit  does  not  amount  to  many  minutes  ;  but  to  make  it 
as  large  as  possible  very  distant  places  are  selected  for  observation. 
Thus  in  the  transit  of  1769,  among  the  places  selected,  two  of  the 
most  favorable  were  Wardhuz  in  Lapland,  and  Oteheite,*  one  of 
the  South  Sea  Islands. 

The  principle  on  which  the  sun's  horizontal  parallax  is  estimated 
from  the  transit  of  Venus,  may  be  illustrated  as  follows :  Let  E 
(Fig.  61,)  be  the  earth,  V  Venus,  and  S  the  sun.  Suppose  A,  B, 
two  spectators  at  opposite  extremities  of  that  diameter  of  the  earth 
which  is  perpendicular  to  the  ecliptic.  The  spectator  at  A  will 
see  Venus  on  the  sun's  disk  at  a,  and  the  spectator  at  B  will  see 
Venus  at  b  ;  and  since  AV  and  BV  may  be  considered  as  equal 
to  each  other,  as  also  V6  and  Va,  therefore  the  triangles  AVB  and 
Vab  are  similar  to  each  other,  and  AV  :  Va : :  AB  :  db.  But  the 
ratio  of  AV  to  Va  is  known,  (Art.  308) ;  hence,  the  ratio  of  AB  to 

*  Now  written  Tahiti. 


188  THE    PLANETS. 

db  is  known,  and  when  the  angular  value  of  ab  as  seen  from  the 
earth,  is  found,  that  of  AB  becomes  known,  as  seen  from  the  sun  ;* 
and  half  AB,  or  the  semi-diameter  of  the  earth  as  seen  from  the 


Fig.  61. 


sun,  is  the  sun's  horizontal  parallax.  To  find  the  apparent  diameter 
of  db,  we  have  only  to  find  the  breadth  of  the  space  between  the 
two  chords.  Now,  we  can  ascertain  the  value  of  each  chord  by 
the  time  occupied  in  describing  it,  since  the  motions  of  Venus  and 
those  of  the  sun  are  accurately  known  from  the  tables.  Each 
chord  being  double  the  sine  of  half  the  arc  cut  off  by  it,  therefore 
the  sine  of  half  the  arc  and  of  course  the  versed  sine  becomes 
known,  and  the  difference  of  the  two  versed  sines  is  the  breadth 
of  the  zone  in  question.  There  are  many  circumstances  to  be 
taken  into  the  account  in  estimating,  from  observations  of  this 
kind,  the  sun's  horizontal  parallax ;  but  the  foregoing  explanation 
may  be  sufficient  to  give  the  learner  an  idea  of  the  general  princi- 
ples of  this  method.  The  appearance  of  Venus  on  the  sun's  disk, 
being  that  of  a  well  defined  black  spot,  and  the  exactness  with 
which  the  moment  of  external  or  internal  contact  may  be  deter- 
mined, are  circumstances  favorable  to  the  exactness  of  the  result ; 
and  astronomers  repose  so  much  confidence  in  the  estimation  of 
the  sun's  horizontal  parallax  as  derived  from  the  observations  on 
the  transit  of  1769,  that  this  important  element  is  thought  to  be 
ascertained  within  TV  of  a  second.  The  general  result  of  all  these 
observations  give  the  sun's  horizontal  parallax  8."6,  or  more  ex- 
actly, 8."5776.f 

*  If,  for  example,  ab  is  2$  times  AB,  (which  is  nearly  the  fact,)  then  if  AB  were  on 
the  sun  instead  of  on  the  earth,  it  would  subtend  an  angle  at  the  eye  equal  to  ^  of  ab. 

But  if  viewed  from  the  sun,  the  distance  being  the  same,  its  apparent  diameter  must  be 
the  same, 
t  Delambre,  t.  2.  Vince's  Complete  Syst.  vol.  1.  Woodhouse,  p.  754.  Herschel,  p.  243, 


SUPERIOR   PLANETS.  189 

320.  During  the  transits  of  Venus  over  the  sun's  disk  in  1761 
and  1769,  a  sort  of  penumbral  light  was  observed  around  the 
planet  by  several  astronomers,  which  was  thought  to  indicate  an 
atmosphere.  This  appearance  was  particularly  observable  while 
the  planet  was  coming  on  and  going  off  the  solar  disk.  The  total 
immersion  and  emersion  were  not  instantaneous ;  but  as  two  drops 
of  water,  when  about  to  separate,  form  a  ligament  between  them, 
so  there  was  a  dark  shade  stretched  out  between  Venus  and  the 
sun,  and  when  the  ligament  broke,  the  planet  seemed  to  have  got 
about  an  eight  part  of  her  diameter  from  the  limb  of  the  sun.* 
The  various  accounts  of  the  two  transits  abound  with  remarks  like 
these,  which  indicate  the  existence  of  an  atmosphere  about  Venus 
of  nearly  the  density  and  extent  of  the  earth's  atmosphere.  Similar 
proofs  of  the  existence  of  an  atmosphere  around  this  planet,  are 
derived  from  appearances  of  twilight. 


CHAPTER  X. 


OF    THE    SUPERIOR    PLANETS MARS,  JUPITER,  SATURN,  AND  URANUS. 

321.  THE  Superior  planets  are  distinguished  from  the  Inferior, 
by  being  seen  at  all  distances  from  the  sun  from  0°  to  180°. 
Having  their  orbits  exterior  to  that  of  the  earth,  they  of  course 
never  come  between  us  and  the  sun,  that  is,  they  never  have  any 
inferior  conjunction  like  Mercury  and  Venus,  but  they  are  some- 
times seen  in  superior  conjunction,  and  sometimes  in  opposition. 
Nor  do  they,  like  the  inferior  planets,  exhibit  to  the  telescope  dif- 
ferent phases,  but,  with  a  single  exception,  they  always  present 
the  side  that  is  turned  towards  the  earth  fully  enlightened.  This 
is  owing  to  their  great  distance  from  the  earth ;  for  were  the  spec- 
tator to  stand  upon  the  sun,  he  would  of  course  always  have  the 
illuminated  side  of  each  of  the  planets  turned  towards  him ;  but, 
so  distant  are  all  the  superior  planets  except  Mars,  that  they  are 

*  Edinb.  Encyc.  Art.  Astronomy. 


190  THE   PLANETS. 

viewed  by  us  very  nearly  in  the  same  manner  as  they  would  be  if 
we  actually  stood  on  the  sun. 

322.  Mars  is  a  small  planet,  his  diameter  being  only  about  half 
that  of  the  earth,  or  4200  miles.  He  also,  at  times,  comes  nearer  to 
the  earth  than  any  other  planet  except  Venus.  His  mean  distance 
from  the  sun  is  142,000,000  miles;  but  his  orbit  is  so  eccentric 
that  his  distance  varies  much  in  different  parts  of  his  revolution. 
Mars  is  always  very  near  the  ecliptic,  never  varying  from  it  2°. 
He  is  distinguished  from  all  the  other  planets  by  his  deep  red  color, 
and  fiery  aspect ;  but  his  brightness  and  apparent  magnitude  vary 
much  at  different  times,  being  sometimes  nearer  to  us  than  at 
others,  by  the  whole  diameter  of  the  earth's  orbit,  that  is,  by  about 
190,000,000  of  miles.  When  Mars  is  on  the  same  side  of  the  sun 
with  the  earth,  or  at  his  opposition,  he  comes  within  47,000,000 
miles  of  the  earth,  and  rising  about  the  time  the  sun  sets  surprises 
us  by  his  magnitude  and  splendor;  but  when  he  passes  to  the 
other  side  of  the  sun  to  his  superior  conjunction,  he  dwindles  to 
the  appearance  of  a  small  star,  being  then  237,000,000  miles  from 
us.  Thus,  let  M  (Fig.  62,)  represent  Mars  in  opposition,  and  M' 

Fig.  62. 


in  the  superior  conjunction.  It  is  obvious  that  in  the  former  situa- 
tion, the  planet  must  be  nearer  to  the  earth  than  in  the  latter  by 
the  whole  diameter  of  the  earth's  orbit. 


JUPITER.  191 

323.  Mars  is  the  only  one  of  the  superior  planets  which  exhibits 
phases.     When  he  is  towards  the  quadratures  at  Q  or  Q',  it  is 
evident  from  the  figure  that  only  a  part  of  the  circle  of  illumina- 
tion is  turned  towards  the  earth,  such  a*  portion  of  the  remoter 
part  of  it  being  concealed  from  our  view  as  to  render  the  form 
more  or  less  gibbous. 

324.  "When  viewed  with  a  powerful  telescope,  the  surface  of 
Mars  appears  diversified  with  numerous   varieties  of  light   and 
shade.     The  region  around  the  poles  is  marked  by  white  spots, 
which  vary  their  appearance  with  the  changes  of  seasons  in  the 
planet.     Hence  Dr.  Herschel  conjectured  that  they  were  owing 
to  ice  and  snow,  which  alternately  accumulates  and  melts,  accord- 
ing to  the  position  of  each  pole  with  respect  to  the  sun.*     It  has 
been  common  to  ascribe  the  ruddy  light  of  this  planet  to  an  exten- 
sive and  dense  atmosphere,  which  was  said  to  be  distinctly  indi- 
cated, by  the  gradual  diminution  of  light  observed  in  a  star  as  it 
approached  very  near  to  the  planet  in  undergoing  an  occultation ; 
but  more  recent  observations  afford  no  such  evidence  of  an  atmo- 
sphere.f 

By  observations  on  the  spots  we  learn  that  Mars  revolves  on  his 
axis  in  very  nearly  the  same  time  with  the  earth,  (24h.  39m.  218.3); 
and  that  the  inclination  of  his  axis  to  that  of  the  ecliptic  is  also 
nearly  the  same,  being  30°  18'  10".8.J 

325.  As  the  diurnal  rotation  of  Mars  is  nearly  the  same  as  that 
of  the  earth,  we  might  expect  a  similar  flattening  at  the  poles, 
giving  to  the  planet  a  spheroidal  figure.     Indeed  the  compression 
or  ellipticity  of  Mars  greatly  exceeds  that  of  the  earth,  being  no 
less  than  TV  of  the  equatorial  diameter,  while  that  of  the  earth  is 
only  aiT,  (Art.  138.)     This  remarkable  flattening  of  the  poles  of 
Mars  has  been  supposed  to  arise  from  a  great  variation  of  density 
in  the  planet  in  different  parts.§ 

326.  JUPITER  is  distinguished  from  all  the  other  planets  by  his 
vast  magnitude.     His  diameter  is  89,000  miles,  and  his  volume 

»  Phil.  Trans.  1784.  t  Sir  James  South,  Phil.  Trans.  1833. 

t  Baily's  Tables,  p.  29.  §  Ed.  Encyc.  Art.  Astronomy. 


192  THE   PLANETS. 

1280  times  that  of  the  earth.  His  figure  is  strikingly  spheroidal, 
the  equatorial  being  larger  than  the  polar  diameter  in  the  propor- 
tion of  107  to  100.  (See  Frontispiece,  Fig.  4.)  Such  a  figure 
might  naturally  be  expected  from  the  rapidity  of  his  diurnal  rota- 
tion, which  is  accomplished  in  about  10  hours.  A  place  on  the 
equator  of  Jupiter  must  turn  27  times  as  fast  as  on  the  terrestrial 
equator.  The  distance  of  Jupiter  from  the  sun  is  nearly  490,000,000 
miles,  and  his  revolution  around  the  sun  occupies  nearly  12 
years. 

327.  The  view  of  Jupiter  through  a  good  telescope,  is  one  of 
the  most  magnificent  and    interesting   spectacles  in   astronomy. 
The  disk  expands  into  a  large  and  bright  orb  like  the  full  moon ; 
the  spheroidal  figure  which  theory  assigns  to  revolving  spheres,  is 
here  palpably  exhibited  to  the  eye ;  across  the  disk,  arranged  in 
parallel  stripes,  are  discerned  several  dusky  bands,  called  belts ; 
and  four  bright  satellites,  always  in  attendance,  but  ever  varying 
their  positions,  compose  a  splendid  retinue.     Indeed,  astronomers 
gaze  with  peculiar  interest  on  Jupiter  and  his  moons  as  affording 
a  miniature  representation  of  the  whole  solar  system,  repeating  on 
a  smaller  scale,  the  same  revolutions,  and  exemplifying,  in  a  man- 
ner more  within  the  compass  of  our  observation,  the  same  laws  as 
regulate  the  entire  assemblage  of  sun  and  planets.     (See  Fig.  63.) 

328.  The  Belts  of  Jupiter,  are  variable  in  their  number  and  di- 
mensions.    With  the  smaller  telescopes,  only  one  or  two  are  sean 
across  the  equatorial  regions  ;  but  with  more  powerful  instruments, 
the  number  is  increased,  covering  a  great  part  of  the  whole  disk. 
Different  opinions  have  been  entertained  by  astronomers  respect- 
ing the  cause  of  the  belts  ;  but  they  have  generally  been  regarded 
as  clouds  formed  in  the  atmosphere  of  the  planet,  agitated  by 
winds,  as  is  indicated  by  their  frequent  changes,  and  made  to  as- 
sume the  form  of  belts  parallel  to  the  equator  by  currents  that  cir- 
culate around  the  planet  like  the  trade  winds  and  other  currents 
that  circulate  around  our  globe.*     Sir  John  Herschel  supposes 
that  the  belts  are  not  ranges  of  clouds,  but  portions  of  the  planet 
itself  brought   into  view   by   the   removal  of  clouds   and  mists 

*  Ed.  Encyc.  Art.  Astronomy. 


JUPITER. 

that  exist  in  the  atmosphere  of  the  planet  through  which  are  open- 
ings made  by  currents  circulating  around  Jupiter.* 

329.  The  Satellites  of  Jupiter  may  be 'seen  with  a  telescope  of 
very  moderate  powers.     Even  a  common  spy-glass  will  enable  us 
to  discern  them.     Indeed  one  or  two  of  them  have  been  occasion- 
ally seen  with  the  naked  eye.     In  the  largest  telescopes,  they  sev- 
erally appear  as  bright  as  Sirius.     With  such  an  instrument  the 
view  of  Jupiter  with  his  moons  and  belts  is  truly  a  magnificent 
spectacle,  a  world  within  itself.     As  the  orbits  of  the  satellites  do 
not  deviate  far  from  the  plane  of  the  ecliptic,  and  but  little  from 
the  equator  of  the  planet,  they  are  usually  seen  in  nearly  a  straight 
line  with  each  other  extending  across  the  central  part  of  the  disk. 
(See  Frontispiece.) 

330.  Jupiter's  satellites  are  distinguished  from  one  another  by 
the  denominations  of  first,  second*  third,  and  fourth,  according  to 
their  relative  distances  from  Jupiter,  the  first  being  that  which  is 
nearest  to  him.     Their  apparent  motion  is  oscillatory,  like  that  of 
a  pendulum,  going  alternately  from  their  greatest  elongation  on 
one  side  to  their  greatest  elongation  on  the  other,  sometimes  in  a 
straight  line,  and  sometimes  in  an  elliptical  curve,  according  to  the 
different  points  of  view  in  which  we  observe  them  from  the  earth. 
They  are  sometimes  stationary ;  their  motion  is  alternately  direct 
and  retrograde  ;  and,  in  short,  they  exhibit  in  miniature  all  the 
phenomena  of  the  planetary  system.     Various  particulars  of  the 
system  are  exhibited  in  the  following  table.     The  distances  are 
given  in  radii  of  the  primary. 


Satellite. 

Diameter. 

Mean  Distance. 

Sidereal  Revolution. 

1 
2 
3 
'      4 

2508 
2068 
3377 
2890 

6.04853 
9.62347 
15.35024 
26.99835 

Id.  18h.  28m. 
3      13      14 
7        3      43 
16      16      32 

Hence  it  appears,  first,  that  Jupiter's  satellites  are  all  somewhat 
larger  than  the  moon,  (except  the  second,  which  is  very  nearly 
the  size  of  the  moon,)  and  the  third  the  largest  of  the  whole,  but  the 

*  Herschel's  Astron.  p.  266 
25 


194  THE    PLANETS. 

diameter  of  the  latter  is  only  about  -fa  part  of  that  of  the  primary ; 
secondly,  that  the  distance  of  the  innermost  satellite  from  the 
planet  is  three  times  his  diameter,  while  that  of  the  outermost 
satellite  is  nearly  fourteen  times  his  diameter ;  thirdly,  that  the 
first  satellite  completes  its  revolution  around  the  primary  in  one 
day  and  three  fourths,  while  the  fourth  satellite  requires  nearly 
sixteen  and  three  fourths  days. 

331.  The  orbits  of  the  satellites  are  nearly  or  quite  circular,  and 
deviate  but  little  from  the  plane  of  the  planet's  equator,  and  of 
course  are  but  slightly  inclined  to  the  plane  of  his  orbit.     They 
are,  therefore,  in  a  similar  situation  with  respect  to  Jupiter  as  the 
moon  would  be  with  respect  to  the  earth  if  her  orbit  nearly  coin- 
cided with  the  ecliptic,  in  which  case  she  would  undergo  an  eclipse 
at  every  opposition. 

332.  The  eclipses  of  Jupiter's  satellites,  in  their  general  concep- 
tion, are  perfectly  analogous  to  those  of  the  moon,  but  in  their  de- 
tail they  differ  in  several  particulars.     Owing  to  the  much  greater 
distance  of  Jupiter  from  the  sun,  and  its  greater  magnitude,  the 
cone  of  its  shadow  is  much  longer  and  larger  than  that  of  the 
earth,  (Art.  246.)     On  this  account,  as  well  as  on  account  of  the 
little  inclination  of  their  orbits  to  that  of  their  primary,  the  three 
inner  satellites  of  Jupiter  pass  through  the  shadow,  and  are  totally 
eclipsed  at  every  revolution.     The  fourth  satellite,  owing  to  the 
greater  inclination  of  its  orbit,  sometimes  though  rarely  escapes 
eclipse,  and  sometimes  merely  grazes  the  limits  of  the  shadow  or 
suffers  a  partial  eclipse.*     These  eclipses,  moreover,  are  not  seen 
as  is  the  case  with  those  of  the  moon,  from  the  center  of  their  mo- 
tion, but  from  a  remote  station,  and  one  whose  situation  with  re- 
spect to  the  line  of  the  shadow  is  variable.     This,  of  course,  makes 
no  difference  in  the  times  of  the  eclipses,  but  a  very  great  one  in 
their  visibility,  and  in  their  apparent  situations  with  respect  to  the 
planet  at  the  moment  of  their  entering  or  quitting  the  shadow. 

333.  The  eclipses  of  Jupiter's  satellites  present  some  curious 
phenomena,  which  will  be  understood  from  the  following  diagram. 

*  Sir  J.  Herschel,  Ast.  p.  276. 


JUPITER.  195 

Let  A,  B,  C,  D,  (Fig.  63,)  represent  the  earth  in  different  parts  of 
its  orbit ;  J,  Jupiter  in  his  orbit  MN,  surrounded  by  his  four  satel- 
lites, the  orbits  of  which  are  marked  1,  2,  3,  4.  At  a  the  first 
satellite  enters  the  shadow  of  the  planet,  and  emerges  from  it  at 
6,  and  advances  to  its  greatest  elongation  at  c.  Since  the  shadow 
is  always  opposite  to  the  sun,  only  the  immersion  of  a  satellite 
will  be  visible  to  the  earth  while  the  earth  is  somewhere  between 

Fig.  63. 


C  and  A,  that  is,  while  the  earth  is  passing  from  the  position 
where  it  has  the  planet  in  superior  conjunction,  to  that  where  it 
has  the  planet  in  opposition  ;  for  while  the  earth  is  in  this  situation, 
the  planet  conceals  from  its  view  the  emersion,  as  is  evident  from 
the  direction  of  the  visual  rayfd.  For  a  similar  reason  the  emer- 
sion only  is  visible  while  the  earth  passes  from  A  to  C,  or  from 
the  opposition  to  the  superior  conjunction.  In  other  words,  when 
the  earth  is  to  the  westward  of  Jupiter,  only  the  immersions  of  a 
satellite  are  visible  ;  when  the  earth  is  to  the  eastward  of  Jupiter, 
only  the  emersions  are  visible.  This,  however,  is  strictly  true  only 
of  the  first  satellite  ;  for  the  third  and  fourth,  and  sometimes  even 
the  second,  owing  to  their  greater  distances  from  Jupiter,  occa- 
sionally disappear  and  reappear  on  the  same  side  of  the  disk. 
The  reason  why  they  should  reappear  on  the  same  side  of  the 
disk,  will  be  understood  from  the  figure.  Conceive  the  whole  sys- 
tem of  Jupiter  and  his  satellites  as  projected  on  the  more  distant 
concave  sphere,  by  lines  drawn,  like/cZ,  from  the  observer  on  the 
earth  at  D  through  the  planet  and  each  of  the  satellites  ;  then  it  is 
evident  that  the  remoter  parts  of  the  shadow  where  the  exterior  sat- 
ellites traverse  it,  will  fall  to  the  westward  of  the  planet,  and  of 


196  THE   PLANETS. 

course  these  satellites  as  they  emerge  from  the  shadow  will  be  pro- 
jected to  a  point  on  the  same  side  of  the  disk  as  the  point  of  their 
immersion.  The  same  mode  of  reasoning  will  show  that  when 
the  earth  is  to  the  eastward  of  the  planet,  the  immersions  and 
emersions  of  the  outermost  satellites  will  be  both  seen  on  the  east 
side  of  the  disk.  When  the  earth  is  in  either  of  the  positions  C 
or  A,  that  is,  at  the  superior  conjunction  or  opposition  of  the  planet, 
both  the  immersions  and  emersions  take  place  behind  the  planet, 
and  the  eclipses  occur  close  to  the  disk. 

334.  When  one  of  the  satellites  is  passing  between  Jupiter  and 
the  sun,  it  casts  a  shadow  upon  its  primary,  which  is  seen  by  the 
telescope  travelling  across  the  disk  of  Jupiter,  as  the  shadow  of  the 
moon  would  be  seen  to  traverse  the  earth  by  a  spectator  favor- 
ably situated  in  space.     When  the  earth  is  to  the  westward  of  Ju- 
piter, as  at  D,  the  shadow  reaches  the  disk  of  the  planet,  or  is  seen 
on  the  disk,  before  the  satellite  itself  reaches  it.     For  the  satellite 
will  not  enter  on  the  disk,  until  it  comes  up  to  the  line  fd  at  d,  a 
point  which  it  reaches  later  than  its  shadow  reaches  the  same  line. 
After  the  earth  has  passed  the  opposition,  as  at  B,  then  the  satel- 
lite will  reach  the  visual  ray  at  d  sooner  than  the  shadow,  and 
of  course  be  sooner  projected  on  the  disk.     In  the  transits  of  Ju- 
piter's satellites,  which  with  very  powerful  telescopes  may  be  ob- 
served with  great  precision,  the  satellite  itself  is  sometimes  seen  on 
the  disk  as  a  bright  spot,  if  it  chances  to  be  projected  upon  one  of 
the  belts.     Occasionally,  also,  it  is  seen  as  a  dark  spot,  of  smaller 
dimensions  than  the  shadow.     This  curious  fact  has  led  to  the 
conclusion,  that  certain  of  the  satellites  have  sometimes  on  their 
own  bodies  or  in  their  atmospheres,  obscure  spots  of  great  extent.* 

335.  A  very  singular  relation  subsists  between  the  mean  motions 
of  the  three  first  satellites  of  Jupiter.     The  mean  longitude  of  thp 
first  satellite,  minus  three  times  that  of  the  second,  plus  twice  that  of 
the  third,  always  equals  180  degrees.     A  curious  consequence  of 
this  relation  is,  that  the  three  satellites  can  never  be  all  eclipsed  at 
the  same  time ;  for  then  their  longitudes  would  be  equal,  and  of 

*  Sir  J.  HerscheL 


JUPITER.  197 

course  the  sum  of  their  longitudes  would  be  nothing.*  It  will  be 
remarked,  that  these  phenomena  are  such  as  would  present  them- 
selves to  a  spectator  on  Jupiter,  and  not  to  a  spectator  on  the 
earth. 

336.  The  eclipses  of  Jupiter's  satellites  have  been  studied  with 
great  attention  by  astronomers,  on  account  of  their  affording  one 
of  the  easiest  methods  of  determining  the  longitude.      On  this 
subject  Sir  J.  Herschel   remarks  :f     The  discovery  of  Jupiter's 
satellites  by  Galileo,  which  was  one  of  the  first  fruits  of  the  inven- 
tion of  the  telescope,  forms  one  of  the  most  memorable  epochs  in 
the  history  of  astronomy.     The  first  astronomical  solution  of  the 
great  problem  of  "  the  longitude," — the  most  important  problem 
for  the  interests  of  mankind  that  has  ever  been  brought  under  the 
dominion  of  strict  scientific  principles,  dates  immediately  from 
their  discovery.     The  final  and  conclusive  establishment  of  the 
Copernican  system  of  astronomy,  may  also  be  considered  as  refer- 
able to  the  discovery  and  study  of  this  exquisite  miniature  system, 
in  which  the  laws  of  the  planetary  motions,  as  ascertained  by 
Kepler,  and  especially  that  which  connects  their  periods  and  dis- 
tances, were  speedily  traced,  and  found  to  be  satisfactorily  main- 
tained. 

337.  The  entrance  of  one  of  Jupiter's  satellites  into  the  shadow 
of  the  primary  being  seen  like  the  entrance  of  the  moon  into  the 
earth's  shadow,   at  the   same  moment  of  absolute  time,  at  all 
places  where  the  planet  is  visible,  and  being  wholly  independent  of 
parallax  ;  being,  moreover,  predicted  beforehand  with  great  accu- 
racy for  the  instant  of  its  occurrence  at  Greenwich,  and  given  in 
the  Nautical  Almanac  ;  this  would  seem  to  be  one  of  those  events 
(Art.  273,)  which  are  peculiarly  adapted  for  finding  the  longitude. 
It  must  be  remarked,  however,  that  the  extinction  of  light  in  the 
satellite  at  its  immersion,  and  the  recovery  of  its  light  at  its  emer- 
sion, are  not  instantaneous,  but  gradual ;  for  the  satellite,  like  the 
moon,   occupies   some  time  in  entering  into  the  shadow  or  in 
emerging  from  it,  which  occasions  a  progressive  diminution  or  in- 

*  Biot,  Ast.  Phy».  t  Elements  of  Ast.  p.  279. 


198  THE    PLANETS. 

crease  of  light.  The  better  the  light  afforded  by  the  telescope 
with  which  the  observation  is  made,  the  later  the  satellite  will  be 
seen  at  its  immersion,  and  the  sooner  at  its  emersion.*  In  noting 
the  eclipses  even  of  the  first  satellite,  the  time  must  be  considered 
as  uncertain  to  the  amount  of  20  or  30  seconds ;  and  those  of  the 
other  satellites  involve  still  greater  uncertainty.  Two  observers, 
in  the  same  room,  observing  with  different  telescopes  the  same 
eclipse,  will  frequently  disagree  in  noting  its  time  to  the  amount 
of  15  or  20  seconds  ;  and  the  difference  will  be  always  the  same 
way.f 

Better  methods,  therefore,  of  finding  the  longitude  are  now 
employed,  although  the  facility  with  which  the  necessary  observa- 
tions can  be  made,  and  the  little  calculation  required,  still  render 
this  method  eligible  in  many  cases  where  extreme  accuracy  is  not 
important.  As  a  telescope  is  essential  for  observing  an  eclipse  of 
one  of  the  satellites,  it  is  obvious  that  this  method  cannot  be  prac- 
ticed at  sea. 

338.  The  grand  discovery  of  the  progressive  motion  of  light, 
was  first  made  by  observations  on  the  eclipses  of  Jupiter's  satel- 
lites.    In  the  year  1675,  it  was  remarked  by  Roemer,  a  Danish 
astronomer,  on  comparing.together  observations  of  these  eclipses 
during  many  successive  years,  that  they  take  place  sooner  by 
about  sixteen  minutes    (16m.   26s. 6)    when   the  earth  is  on  the 
same  side  of  the  sun  with  the  planet,  than  when  she  is  on  the  op- 
posite side.     This  difference  he  ascribed  to  the  progressive  motion 
of  light,  which  takes  that  time  to  pass  through  the  diameter  of  the 
earth's  orbit,  making  the  velocity  of  light  about  192,000  miles  per 
second.     So  great  a  velocity  startled  astronomers  at  first,  and  pro- 
duced some  degree  of  distrust  of  this  explanation  of  the  phenome- 
non ;  but  the  subsequent  discovery  of  the  aberration  of  light  (Art. 
195,)  led  to  an  independent  estimation  of  the  velocity  of  light 
with  almost  precisely  the  same  result. 

339.  SATURN  comes  next  in  the  series  as  we  recede  from  the 

*  This  is  the  reason  why  observers  are  directed  in  the  Nautical  Almanac  to  use  tele, 
•copes  of  a  certain  power. 
t  Woodhouse,  p.  840. 


SATURN.  199 

sun,  and  has,  like  Jupiter,  a  system  within  itself,  on  a  scale  of  great 
magnificence.  In  size  it  is,  next  to  Jupiter,  the  largest  of  the 
planets,  being  79,000  miles  in  diameter,  or  about  1,000  times  as 
large  as  the  earth.  It  has  likewise  belts  on  its  surface  and  is  at- 
tended by  seven  satellites.  But  a  still  more  wonderful  appendage 
is  its  Ring,  a  broad  wheel  encompassing  the  planet  at  a  great  dis- 
tance from  it.  We  have  already  intimated  that  Saturn's  system 
is  on  a  grand  scale.  As,  however,  Saturn  is  distant  from  us  nearly 
900,000,000  miles,  we  are  unable  to  obtain  the  same  clear  and 
striking  views  of  his  phenomena  as  we  do  of  the  phenomena  of 
Jupiter,  although  they  really  present  a  more  wonderful  mechanism. 

340.  Saturn's  ring,  when  viewed  with  telescopes  of  a  high 
power,  is  found  to  consist  of  two  concentric  rings,*  separated  from 
each  other  by  a  dark  space.  (See  Frontispiece.)  Although  this 
division  of  the  rings  appears  to  us,  on  account  of  our  immense  dis- 
tance, as  only  a  fine  line,  yet  it  is  in  reality  an  interval  of  not  less 
than  about  1800  miles.  The  dimensions  of  the  whole  system  are 
in  round  numbers,  as  follows  :f 

Miles. 

Diameter  of  the  planet,  ....  79,000 
From  the  surface  of  the  planet  to  the  inner  ring,  20,000 
Breadth  of  the  inner  ring,  :  .  .  .  .  17,000 
Interval  between  the  rings,  ..  .  .  1,800 

Breadth  of  the  outer  ring,  ....  10,500 
Extreme  dimensions  from  outside  to  outside,  176,000 
The  figure  represents  Saturn  as  it  appears  to  a  powerful  tele- 
scope, surrounded  by  its  rings,  and  having  its  body  striped  with 
dark  belts,  somewhat  similar  but  broader  and  less  strongly  marked 
than  those  of  Jupiter,  and  owing  doubtless  to  a  similar  cause. 
That  the  ring  is  a  solid  opake  substance,  is  shown  by  its  throwing 
its  shadow  on  the  body  of  the  planet  on  the  side  nearest  the  sun 
and  on  the  other  side  receiving  that  of  the  body.  From  the  par- 
allelism of  the  belts  with  the  plane  of  the  ring,  it  may  be  conjec- 
tured that  the  axis  of  rotation  of  the  planet  is  perpendicular  to 

*  It  is  said  that  several  additional  divisions  of  the  ring  have  been  detected. — (Kater, 
Ast.  Trans,  iv.  383.)  t  Prof.  Struve,  Mem.  Ast.  Soc.,  3.  301. 


200  THE   PLANETS. 

th  it  plane ;  and  this  conjecture  is  confirmed  by  the  occasional 
a]  pearance  of  extensive  dusky  spots  on  its  surface,  which  when 
watched  indicate  a  rotation  parallel  to  the  ring  in  lOh.  29m.  17s. 
This  motion,  it  will  be  remarked,  is  nearly  the  same  with  the  diur- 
nal motion  of  Jupiter,  subjecting  places  on  the  equator  of  the 
planet  to  a  very  swift  revolution,  and  occasioning  a  high  degree 
of  compression  at  the  poles,  the  equatorial  being  to  the  polar  di- 
ameter in  the  high  ratio  of  11  to  10.  But  it  is  remarkable  that  the 
globe  of  Saturn  appears  to  be  flattened  at  the  equator  as  well  as 
at  the  poles.  The  polar  compression  extends  to  a  great  distance 
over  the  surface  of  the  planet,  and  the  greatest  diameter  is  that  of 
the  parallel  of  43°  of  latitude.  The  dis'k  of  Saturn,  therefore,  re- 
sembles a  square  of  which  the  four  corners  have  been  rounded  off.* 
It  requires  a  telescope  of  high  magnifying  powers  and  a  strong 
light  to  give  a  full  and  striking  view  of  Saturn  with  his  rings  and 
belts  and  satellites  ;  for  we  must  bear  in  mind  that  at  the  distance 
of  Saturn  one  second  of  angular  measurement  corresponds  to  4,000 
miles,  a  space  equal  to  the  semi-diameter  of  our  globe.  But  with 
a  telescope  of  moderate  powers,  the  leading  phenomena  of  the 
ring  itself  may  be  observed. 

341.  Saturn1  s  ring,  in  its  revolution  around  the  sun,  always  re- 
mains parallel  to  itself. 

If  we  hoM  opposite  to>  the  eye  a  circular  ring  or  disk  like  a 
piece  of  coin,  it  will  appear  as  a  complete  circle  when  it  is  at  right 
angles  to  the  axis  of  vision,  but  when  oblique  to  that  axis  it  will 
be  projected  into  an  ellipse  more  and  more  flattened  as  its  obliquity 
is  increased,  until,  when  its  plane  coincides  with  the  axis  of  vision, 
it  is  projected  into  a  straight  line.  Let  us  place  on  the  table  a 
lamp  to  represent  the  sun,  and  holding  the  ring  at  a  certain  dis- 
tance inclined  a  little  towards  the  lamp,  let  us  carry  it  round  the 
lamp,  always  keeping  it  parallel  to  itself.  During  its  revolution  it 
will  twice  present  its  edge  to  the  lamp  at  opposite  points,  and 
twice  at  places  90°  distant  from  those  points,  it  will  present  its 
broadest  face  towards  the  lamp.  At  intermediate  points,  it  will 
exhibit  an  ellipse  more  or  less  open,  according  as  it  is  nearer  one 

*  Sir  W.  Herschel,  Phil.  Tr.  1806,  Part  II. 


SATURN.  201 

or  the  other  of  the  preceding  positions.  It  will  be  seen  also  that 
in  one  half  of  the  revolution  the  lamp  shines  on  one  side  of  the 
ring,  and  in  the  other  half  of  the  revolution  on  the  other  side. 
Such  would  be  the  successive  appearances  of  Saturn's  ring  to  a 
spectator  on  the  sun ;  and  since  the  earth  is,  in  respect  to  so  dis- 
tant a  body  as  Saturn,  very  near  the  sun,  those  appearances  are 
presented  to  us  in  nearly  the  same  manner  as  though  we  viewed 
them  from  the  sun.  Accordingly,  we  sometimes  see  Saturn's  ring 
under  the  form  of  a  broad  ellipse,  which  grows  continually  more 
and  more  acute  until  it  passes  into  a  line,  and  we  either  lose  sight 
of  it  altogether,  or  with  the  aid  of  the  most  powerful  telescopes, 
we  see  it  as  a  fine  thread  of  light  drawn  across  the  disk  and  pro- 
jecting out  from  it  on  each  side.  As  the  whole  revolution  occupies 
30  years,  and  the  edge  is  presented  to  the  sun  twice  in  the  revolu- 
tion, this  last  phenomenon,  namely,  the  disappearance  of  the  ring, 
takes  place  every  15  years. 

342.  The  learner  may  perhaps  gain  a  clearer  idea  of  the  fore- 
going appearances  from  the  following  diagram : 

Let  A,  B,  C,  &c.  represent  successive  positions  of  Saturn  and 
his  ring  in  different  parts  of  his  orbit,  while  ab  represents  the 
orbit  of  the  earth.*  Were  the  ring  when  at  C  and  G  perpendicu- 

Fig.  64. 


lar  to  the  line  CG,  it  would  be  seen  by  a  spectator  situated  at  a 
or  b  a  perfect  circle,  but  being  inclined  to  the  line  of  vision  28°  4', 

*.It  may  be  remarked  by  the  learner,  that  these  orbits  are  made  so  elliptical,  not  to 
represent  the  eccentricity  of  either  the  earth's  or  Saturn's  orbit,  but  merely  as  the  pro- 
jection  of  circles  seen  very  obliquely. 

26 


202  THE   PLANETS. 

it  is  projected  into  an  ellipse.  This  ellipse  contracts  in  breadth 
as  the  ring  passes  towards  its  nodes  at  A  and  E,  where  it  dwindles 
into  a  straight  line.  From  E  to  G  the  ring  opens  again,  becomes 
broadest  at  G,  and  again  contracts  till  it  becomes  a  straight  line  at 
A,  and  from  this  point  expands  till  it  recovers  its  original  breadth 
at  C.  These  successive  appearances  are  all  exhibited  to  a  telescope 
of  moderate  powers.  The  ring  is  extremely  thin,  since  the  small- 
est satellite,  when  projected  on  it,  more  than  covers  it.  The  thick- 
ness is  estimated  at  100  miles. 

343.  Saturn9  s  ring  shines  wholly  by  reflected  light  derived  from 
the  sun.     This  is  evident  from  the  fact,  that  that  side  only  which 
is  turned  towards  the  sun  is  enlightened  ;  and  it  is  remarkable, 
that  the  illumination  of  the  ring  is  greater  than  that  of  the  planet 
itself,  but  the  outer  ring  is  less  bright  than  the  inner.     Although,  as 
we  have  already  remarked,  we  view  Saturn's  ring  nearly  as  though 
we  saw  it  from  the  sun,  yet  the  plane  of  the  ring  produced  may 
pass  between  the  earth  and  the  sun,  in  which  case  also  the  ring 
becomes  invisible,  the  illuminated  side  being  wholly  turned  from 
us.     Thus,  when  the  ring  is  approaching  its  node  atE,  a  spectator 
at  a  would  have,  the  dark  side  of  the  ring  presented  to  him.     The 
ring  was  invisible  in  1833,  and  will  be  invisible  again  in  1847. 
At  present  (1841)  it  is  the  northern  side  of  the  ring  that  is  seeij, 
but  in  1855  the  southern  side  will  come  into  view. 

It  appears,  therefore,  that  there  are  three  causes  for  the  disap- 
pearance of  Saturn's  ring ;  first,  when  the  edge  of  the  ring  is  pre- 
sented to  the  sun ;  secondly,  when  the  edge  is  presented  to  the 
earth  ;  and  thirdly,  when  the  unilluminated  side  is  towards  the 
earth. 

344.  Saturn's  ring  revolves  in  its  own  plane  in  about  10£  hours, 
(lOh.  32m.  15s. 4).     La  Place  inferred  this  from  the  doctrine  of 
universal  gravitation.     He  proved  that  such  a  rotation  was  neces- 
sary, otherwise  the  matter  of  which  the  ring  is  composed  would 
be  precipitated  upon  its  primary.     He  showed  that  in  order  to 
sustain  itself,  its  period  of  rotation  must  be  equal  to  the  time  of 
revolution  of  a  satellite,  circulating  around  Saturn  at  a  distance 
from  it  equal  to  that  of  the  middle  of  the  ring,  which  period  would 


SATURN.  203 

be  about  10?  hours.  By  means  of  spots  in  the  ring  Dr.  Hersche. 
followed  the  ring  in  its  rotation,  and  actually  found  its  period  to 
be  the  same  as  assigned  by  La  Place, — a  coincidence  which  beau- 
tifully exemplifies  the  harmony  of  truth.*' 

345.  Although  the  rings  are  very  nearly  concentric,  yet  recent 
measurements  of  extreme  delicacy  have  demonstrated,  that  the 
coincidence  is  not  mathematically  exact,  but  that  the  center  of 
gravity  of  the  rings  describes  around  that  of  the  body  a  very 
minute  orbit.  This  fact,  unimportant  as  it  may  seem,  is  of  the 
utmost  consequence  to  the  stability  of  the  system  of  rings.  Sup- 
posing them  mathematically  perfect  in  their  circular  form,  and 
exactly  concentric  with  the  planet,  it  is  demonstrable  that  they 
would  form  (in  spite  of  their  centrifugal  force)  a  system  in  a  state 
of  unstable  equilibrium,  which  the  slightest  external  power  would 
subvert — not  by  causing  a  rupture  in  the  substance  of  the  rings — 
but  by  precipitating  them  unbroken  on  the  surface  of  the  planet. •(• 
The  ring  may  be  supposed  of  an  unequal  breadth  in  its  different 
parts,  and  as  consisting  of  irregular  solids,  whose  common  center 
of  gravity  does  not  coincide  with  the  center  of  the  figure.  Were 
it  not  for  this  distribution  of  matter,  its  equilibrium  would  be  de- 
stroyed by  the  slightest  force,  such  as  the  attraction  of  a  satellite, 
and  the  ring  would  finally  precipitate  itself  upon  the  planet.  J 

As  the  smallest  difference  of  velocity  between  the  planet  and 
its  rings  must  infallibly  precipitate  the  rings  upon  the  planet, 
never  more  to  separate,  it  follows  either  that  their  motions  in  their 
common  orbit  round  the  sun,  must  have  been  adjusted  to  each 
other  by  an  external  power,  with  the  minutest  precision,  or  that 
the  rings  must  have  been  formed  about  the  planet  while  subject 
to  their  common  orbitual  motion,  and  under  the  full  and  free  in- 
fluence of  all  the  acting  forces. 

The  rings  of  Saturn  must  present  a  magnificent  spectacle  from 
those  regions  of  the  planet  which  lie  on  their  enlightened  sides, 
appearing  as  vast  arches  spanning  the  sky  from  horizon  to  hori- 
zon, and  holding  an  invariable  situation  among  the  stars.  On 
the  other  hand,  in  the  region  beneath  the  dark  side,  a  solar  eclipse 

•  Systeme  du  Monde,  1.  iv.  c.  8.  t  Sir  J.  Herschel.  t  La  Place. 


204  THE  PLANETS. 

of  15  years  in  duration,  under  their  shadow,  must  afford  (to  our 
ideas)  an  inhospitable  abode  to  animated  beings,  but  ill  compen- 
sated by  the  full  light  of  its  satellites.  But  we  shall  do  wrong 
to  judge  of  the  fitness  or  unfitness  of  their  condition  from  what 
we  see  around  us,  when,  perhaps,  the  very  combinations  which 
convey  to  our  minds  only  images  of  horror,  may  be  in  reality 
theatres  of  the  most  striking  and  glorious  displays  of  beneficent 
contrivance.* 

346.  Saturn  is  attended  by  seven  satellites.     Although  bodies 
of  considerable  size,  their  great  distance  prevents  their  being  vis- 
ible to  any  telescopes  but  such  as  afford  a  strong  light  and  high 
magnifying  powers.     The  outermost  satellite  is  distant  from  the 
planet  more  than  30  times  the  planet's  diameter,  and  is  by  far 
the  largest  of  the  whole.     It  is  the  only  one  of  the  series  whose 
theory  has  been  investigated  further  than  suffices  to  verify  Kep- 
ler's law  of  the  periodic  times,  which  is  found  to  hold  good  here 
as  well  as  in  the  system  of  Jupiter.     It  exhibits,  like  the  satellites 
of  Jupiter,  periodic  variations  of  light,  which  prove  its  revolution 
on  its  axis  in  the  time  of  a  sidereal  revolution   about  Saturn. 
The  next  satellite  in  order,  proceeding  inwards,  is  tolerably  con- 
spicuous ;  the  three  next  are  very  minute,  and  require  pretty  pow- 
erful telescopes  to  see  them ;  while  the  two  interior  satellites, 
which  just  skirt  the  edge  of  the  ring,  and  move  exactly  in  its 
plane,  have  never  been  discovered  but  with  the  most  powerful 
telescopes  which  human  art  has  yet  constructed,  and  then  only 
under  peculiar  circumstances.     At  the  time  of  the  disappearance 
of  the  rings  (to  ordinary  telescopes)  they  were  seen  by  Sir  Wil- 
liam Herschel  with  his  great  telescope,  projected  along  the  edge 
of  the  ring,  and  threading  like  beads  the  thin  fibre  of  light  to  which 
the  ring  is  then  reduced.     Owing  to  the  obliquity  of  the  ring,  and 
of  the  orbits  of  the  satellites  to  that  of  their  primary,  there  are  no 
eclipses  of  the  satellites,  the  two  interior  ones  excepted,  until  near 
the  time  when  the  ring  is  seen  edgewise.f 

347.  URANUS  is  the  remotest  planet  belonging  to  our  system, 

*  Sir  J.  Herschel.  t  Sir  J.  Herschel. 


SATBRN.  205 

*j 

and  is  rarely  visible  except  to  the  telescope.  Although  his  diam- 
eter is  more  than  four  times  that  of  the  earth,  (35,112  miles,)  yet 
his  distance  from  the  sun  is  likewise  nineteen  times  as  great  as 
the  earth's  distance,  or  about  1,800,000,000  miles.  His  revolution 
around  the  sun  occupies  nearly  84  years,  so  that  his  position  in 
the  heavens  for  several  years  in  succession  is  nearly  stationary. 
His  path  lies  very  nearly  in  the  ecliptic,  being  inclined  to  it  less 
than  one  degree,  (46'  28".44.) 

The  sun  himself  when  seen  from  Uranus  dwindles  almost  to  a 
star,  subtending  as  it  does  an  angle  of  only  1'  40"  ;  so  that  the 
surface  of  the  sun  would  appear  there  400  times  less  than  it  does 
to  us. 

This  planet  was  discovered  by  Sir  William  Herschel  on  the 
13th  of  March,  1781.  His  attention  was  attracted  to  it  by  the 
largeness  of  its  disk  in  the  telescope ;  and  finding  that  it  shifted 
its  place  among  the  stars,  he  at  first  took  it  for  a  comet,  but  soon 
perceived  that  its  orbit  was  not  eccentric  like  the  orbits  of  comets, 
but  nearly  circular  like  those  of  the  planets.  It  was  then  recog- 
nized as  a  new  member  of  the  planetary  system,  a  conclusion 
which  has  been  justified  by  all  succeeding  observations. 

348.  Uranus  is  attended  by  six  satellites.  So  minute  objects 
are  they  that  they  can  be  seen  only  by  powerful  telescopes.  In- 
deed the  existence  of  more  than  two  is  still  considered  as  some- 
what doubtful.*  These  two,  however,  offer  remarkable,  and  in- 
deed quite  unexpected  and  unexampled  peculiarities.  Contrary 
to  the  unbroken  analogy  of  the  whole  planetary  system,  the  planes 
of  their  orbits  are  nearly  perpendicular  to  the  ecliptic,  being  inclined 
no  less  than  78°  58'  to  that  plane,  and  in  these  orbits  their  motions 
are  retrograde ;  that  is,  instead  of  advancing  from  west  to  east 
around  their  primary,  as  is  the  case  with  all  the  other  planets  and 
satellites,  they  move  in  the  opposite  direction.f  With  this  excep- 
tion, all  the  'motions  of  the  planets,  whether  around  their  own  axes, 
or  around  the  sun,  are  from  west  to  east. 


*  A  third  satellite  of  Uranus,  is  said  to  have  been  recently  seen  at  Munich.    (Jouu 
Franklin  Inst.  xxiii,  29.) 
t  Sir  J.  Herschel. 


206  THE  PLANETS. 

OF    THE    NEW    PLANETS,    CERES,    PALLAS,    JUNO,    AND    VESTA. 

349.  THE  commencement  of  the  present  century  was  rendered 
memorable  in  the  annals  of  astronomy,  by  the  discovery  of  four 
new  planets  between   Mars   and   Jupiter.     Kepler,  from   some 
analogy  which  he  found  to  subsist  among  the  distances  of  the 
planets  from  the  sun,  had  long  before  suspected  the  existence  of 
one  at  this  distance  ;  and  his  conjecture  was  rendered  more  prob- 
able by  the  discovery  of  Uranus,  which  follows  the  analogy  oi 
the  other  planets.     So  strongly,  indeed,  were   astronomers  im- 
pressed with  the  idea  that  a  planet  would  be  found  between  Mars 
and  Jupiter,  that  in  the  hope  of  discovering  it,  an  association  was 
formed  on  the  continent  of  Europe  of  twenty-four  observers,  who 
divided  the  sky  into  as  many  zones,  one  of  which  was  allotted  to 
each  member  of  the  association.     The  discovery  of  the  first  of 
these  bodies  was  however  made  accidentally  by  Piazzi,  an  astron- 
omer of  Palermo,  on  the  first  of  January,  1801.     It  was  shortly 
afterwards  lost  sight  of  on  account  of  its  proximity  to  the  sun, 
and'  was  not  seen  again  until  the  close  of  the  year,  when  it  was 
re-discovered  in  Germany.     Piazzi  called  it  Ceres  in  honor  of  the 
tutelary  goddess  of  Sicily,  and  her  emblem,  the  sickle  ? ,  has  been 
adopted  as  its  appropriate  symbol. 

The  difficulty  of  finding  Ceres  induced  Dr.  Olbers,  of  Bremen, 
to  examine  with  particular  care  all  the  small  stars  that  lie  near 
her  path,  as  seen  from  the  earth  ;  and  while  prosecuting  these 
observations,  in  March,  1802,  he  discovered  another  similar  body, 
very  nearly  at  the  same  distance  from  the  sun,  and  resembling  the 
former  in  many  other  particulars.  The  discoverer  gave  to  this 
second  planet  the  name  of  Pallas,  choosing  for  its  symbol  the 
lance  $ ,  the  characteristic  of  Minerva. 

350.  The   most   surprising   circumstance  connected  with  the 
discovery  of  Pallas,  was  the  existence  of  two  planets  at  nearly  the 
same  distance  from  the  sun,  and  apparently  having  a  common  node. 
On  account  of  this  singularity,  Dr.  Olbers  was  led  to  conjecture 
that  Ceres  and  Pallas  are  only  fragments  of  a  larger  planet,  which 
had  formerly  circulated  at  the  same  distance,  and  been  shattered 
by  some  internal  convulsion.     La  Grange,  a  mathematician  of  the 


NEW    PLANETS.  207 

first  eminence,  investigated  the  forces  that  would  be  necessary  to 
detach  a  fragment  from  a  planet  with  a  velocity  that  would  cause 
it  to  describe  such  orbits  as  these  bodies  are  found  to  have.  The 
hypothesis  suggested  the  probability  that  there  might  be  other 
fragments,  whose  orbits,  however  they  might  differ  in  eccentricity 
and  inclination,  might  be  expected  to  cross  the  ecliptic  at  a  com- 
mon point,  or  to  have  the  same  node.  Dr.  Olbers,  therefore,  pro- 
posed to  examine  carefully  every  month  the  two  opposite  parts 
of  the  heavens  in  which  the  orbits  of  Ceres  and  Pallas  intersect 
one  another,  with  a  view  to  the  discovery  of  other  planets,  which 
might  be  sought  for  in  those  parts  with  greater  chance  of  success 
than  in  a  wider  zone,  embracing  the  entire  limits  of  these  orbits. 
Accordingly,  in  1804,  near  one  of  the  nodes  of  Ceres  and  Pallas, 
a  third  planet  was  discovered.  This  was  called  Juno,  and  the 
character  $  was  adopted  for  its  symbol,  representing  the  starry 
sceptre  of  the  queen  of  Olympus.  Pursuing  the  same  researches, 
in  1807,  a  fourth  planet  was  discovered,  to  which  was  given  the 
name  of  Vesta,  and  for  its  symbol  the  character  fi  was  chosen, 
an  altar  surmounted  with  a  censer  holding  the  sacred  fire. 

After  this  historical  sketch,  it  will  be  sufficient  to  classify  under 
a  few  heads  the  most  interesting  particulars  relating  to  the  New 
Planets. 

351.  The  average  distance  of  these  bodies  from  the  sun  is 
261,000,000  miles ;  and  it  is  remarkable  that  their  orbits  are  very 
near  together.  Taking  the  distance  of  the  earth  from  the  sun  for 
unity,  their  respective  distances  are  2.77,  2.77,  2.67,  2.37. 

As  they  are  found  to  be  governed,  like  the  other  members  of 
the  solar  system,  by  Kepler's  law,  that  regulates  the  distances  and 
times  of  revolution,  their  periodical  times  are  of  course  pretty 
nearly  equal,  averaging  about  4£  years. 

In  respect  to  the  inclination  of  their  orbits,  there  is  considerable 
diversity.  The  orbit  of  Vesta  is  inclined  to  the  ecliptic  only 
about  7°,  while  that  of  Pallas  is  more  than  34°.  They  all  there- 
fore have  a  higher  inclination  than  the  orbits  of  the  old  planets, 
and  of  course  make  excursions  from  the  ecliptic  beyond  the  limits 
of  the  Zodiac. 

The  eccentricity  of  their  orbits  is  also,  in  general,  greater  than 


208  THE  PLANETS. 

that  of  the  old  planets  ;  and  the  eccentricities  of  the  orbits  of  Pal- 
las and  Juno  exceed  that  of  the  orbit  of  Mercury. 

Their  small  size  constitutes  one  of  their  most  remarkable  pecu- 
liarities. The  difficulty  of  estimating  the  apparent  diameter  of 
bodies  at  once  so  very  small  and  so  far  off,  would  lead  us  to  ex- 
pect different  results  in  the  actual  estimates.  Accordingly,  while 
Dr.  Herschel  estimates  the  diameter  of  Pallas  at  only  80  miles, 
Schroeter  places  it  as  high  as  2,000  miles,  or  about  the  size 
of  the  moon.  The  volume  of  Vesta  is  estimated  at  only  one  fif- 
teen thousandth  part  of  the  earth's,  and  her  surface  is  only  about 
equal  to  that  of  the.  kingdom  of  Spain.*  These  little  bodies  are 
surrounded  by  atmospheres  of  great  extent,  some  of  which  are  un- 
commonly luminous,  and  others  appear  to  consist  of  nebulous  mat- 
ter. These  planets  in  general  shine  with  a  more  vivid  light  than 
might  be  expected  from  their  great  distance  and  diminutive  size 


CHAPTER    XI. 

-  MOTIONS    OF   THE    PLANETARY    SYSTEM. 

352.  WE  have  waited  until  the  learner  may  be  supposed  to  be 
familiar  with  the  contemplation  of  the  heavenly  bodies,  individu- 
ally, before  inviting  his  attention  to  a  systematic  view  of  the 
planets,  and  of  their  motions  around  the  sun.  The  time  has  now 
arrived  for  entering  more  advantageously  upon  this  subject,  than 
could  have  been  done  at  an  earlier  period. 

There  are  two  methods  of  arriving  at  a  knowledge  of  the  mo- 
tions of  the  heavenly  bodies.  One  is  to  begin  with  the  apparent, 
and  from  these  to  deduce  the  real  motions  ;  the  other  is,  to  begin 
with  considering  things  as  they  really  are  in  nature,  and  then  to 
inquire  why  they  appear  as  they  do.  The  latter  of  these  methods 
is  by  far  the  more  eligible ;  it  is  much  easier  than  the  other,  and 

*  New  Encyc.  Brit.,  Art.  Astronomy. 


MOTIONS  OF  THE  PLANETARY  SYSTEM.  209 

proceeding  from  the  less  difficult  to  that  which  is  more  so,  from 
motions  that  are  very  simple  to  such  as  are  complicated,  it- finally 
puts  the  learner  in  possession  of  the  whole  machinery  of  the  heav- 
ens. We  shall,  in  the  first  place,  therefore,  endeavor  to  introduce 
the  student  to  an  acquaintance  with  the  simplest  motions  of  the 
planetary  system,  and  afterwards  to  conduct  him  gradually 
through  such  as  are  more  complicated  and  difficult. 

353.  Let  us  first  of  all  endeavor  to  acquire  an  adequate  idea  of 
absolute  space,  such  as  existed  before  the  creation  of  the  world. 
We  shall  find  it  no  easy  matter  to  form  a  correct  notion  of  infinite 
space  ;  but  let  us  fix  our  attention,  for  some  time,  upon  extension 
alone,  devoid  of  every  thing  material,  without  light  or  life,  and 
without  bounds.     Of  such  a  space  we  could  not  predicate  the 
ideas  of  up  or  down,  east,  west,  north,  or  south,  but  all  reference 
to  our  own  horizon  (which  habit  is  the  most  difficult  of  all  to 
eradicate  from  the  mind)  must  be  completely  set  aside.     Into  such 
a  void  we  would  introduce  the  SUN.     We  would  contemplate  this 
body  alone,  in  the  midst  of  boundless  space,  and  continue  to  fix 
he  attention  upon  this  object,  until  we  had  fully  settled  its  rela- 
tions to  the  surrounding  void.     The  ideas  of  up  and  down  would 
now  present  themselves,  but  as  yet  there  would  be  nothing  to  sug- 
gest any  notion  of  the  cardinal  points.     We  suppose  ourselves 
next  to  be  placed  on  the  surface  of  the  sun,  and  the  firmament  of 
stars  to  be  lighted  up.     The  slow  revolution  of  the  sun  on  his  axis, 
would  be  indicated  by  a  corresponding  movement  of  the  stars  in 
the  opposite  direction ;  and  in  a  period  equal  to  more  than  27  of 
our  days,  the  spectator  would  see  the  heavens  perform  a  complete 
revolution  around  the  sun,  as  he  now  sees  them  revolve  around 
the  earth  once  in  24  hours.     The  point  of  the  firmament  where 
no  motion  appeared,  would  indicate  the  position  of  one  of  the 
poles,  which  being  called  North,  the  other  cardinal  points  would 
be  immediately  suggested. 

Thus  prepared,  we  may  now  enter  upon  the  consideration  of 
the  planetary  motions. 

354.  Standing  on  the  sun,  we  see  all  the  planets  moving  slowly 
around  the  celestial  sphere,  nearly  in  the  same  great  high  way,  and 

27 


210  THE   PLANETS. 

in  the  same  direction  from  west  to  east.  They  move,  however, 
with  very  unequal  velocities.  Mercury  makes  very  perceptible 
progress  from  night  to  night,  like  the  moon  revolving  about  the 
earth,  his  daily  progress  eastward  being  about  one  third  as  great  as 
that  of  the  moon,  since  he  completes  his  entire  revolution  in  about 
three  months.  If  we  watch  the  course  of  this  planet  from  night 
to  night,  we  observe  it,  in  its  revolution,  to  cross  the  ecliptic  in 
two  opposite  points  of  the  heavens,  and  wander  about  7°  from 
that  plane  at  its  greatest  distance  from  it.  Knowing  the  position 
of  the  orbit  of  Mercury  with  respect  to  the  ecliptic,  we  may  now, 
in  imagination,  represent  that  orbit  by  a  great  circle  passing 
through  the  center  of  the  planet  and  the  center  of  the  sun,  and 
cutting  the  plane  of  the  ecliptic  in  two  opposite  points  at  an  angle 
of  7°.  We  may  imagine  the  intersection  of  these  two  great  cir- 
cles, with  the  celestial  vault  to  be  marked  out  in  plain  and  palpable 
lines  on  the  face  of  the  sky  ;  but  we  must  bear  in  mind  that  these 
orbits  are  mere  mathematical  planes,  having  no  permanent  exist- 
ence in  nature,  any  more  than  the  path  of  an  eagle  flying  through 
the  sky  ;  and  if  we  conceive  of  their  orbits  as  marked  on  the  ce- 
lestial vault,  we  must  be  careful  to  attach  to  the  representation 
the  same  notion  as  to  a  thread  or  wire  carried  round  to  trace  out 
the  course  pursued  by  a  horse  in  a  race-ground.* 

The  planes  of  both  the  ecliptic  and  the  orbit  of  Mercury,  may 
be  conceived  of  as  indefinitely  extended  to  a  great  distance  until 
they  meet  the  sphere  of  the  stars ;  but  the  lines  which  the  earth 
and  Mercury  describe  in  those  planes,  that  is,  their  orbits,  may  be 
conceived  of  as  comparatively  near  to  the  sun.  Could  we  now  for 
a  moment  be  permitted  to  imagine  that  the  planes  of  the  ecliptic, 
and  of  the  orbit  of  Mercury,  were  made  of  thin  plates  of  glass, 
and  that  the  paths  of  the  respective  planets  were  marked  out  on 
their  planes  in  distinct  lines,  we  should  perceive  the  orbit  of  the 
earth  to  be  almost  a  perfect  circle,  while  that  of  Mercury  would 


*  It  would  seem  superfluous  to  caution  the  reader  on  so  plain  a  point,  did  not  the 
experience  of  the  instructor  constantly  show  that  young  learners,  from  the  habit  of 
seeing  the  celestial  motions  represented  in  orreries  and  diagrams,  almost  always  fall 
into  the  absurd  notion  of  considering  the  orbits  of  the  planets  as  having  a  distinct  and 
independent  existence. 


MOTIONS  OF  THE  PLANETARY  SYSTEM.  211 

. 

appear  distinctly  elliptical.  But  having  once  made  use  of  a  palpa- 
ble surface  and  visible  lines  to  aid  us  in  giving  position  and  figure 
to  the  planetary  orbits,  let  us  now  throw  aside  these  devices,  and 
hereafter  conceive  of  these  planes  and  orbits  as  they  are  in  nature, 
and  learn  to  refer  a  body  to  a  mere  mathematical  plane,  and  to 
trace  its  path  in  that  plane  through  absolute  space. 

355.  A  clear  understanding  of  the  motions  of  Mercury  and  of 
the  relation  of  its  orbit  to  the  plane  of  the  ecliptic,  will  render  it 
easy  to  understand  the  same  particulars  in  regard  to  each  of  the 
other  planets.     Standing  on  the  sun  we  should  see  each  of  the 
planets  pursuing  a  similar  course  to  that  of  Mercury,  all  moving 
from  west  to  east,  with  motions  differing  from  each  other  chiefly 
in  two  respects,  namely,  in  their  velocities,  and  in  the  distances 
to  which  they  ever  recede  from  the  ecliptic. 

The  earth  revolves  about  the  sun  very  much  like  Venus,  and  to 
a  spectator  on  the  sun,  the  motions  of  these  two  planets  would 
exhibit  much  the  same  appearances.  "We  have  supposed  the  ob- 
server to  select  the  plane  of  the  earth's  orbit  as  his  standard  of 
reference,  and  to  see  how  each  of  the  other  orbits  is  related  to  it ; 
but  such  a  selection  of  the  ecliptic  is  entirely  arbitrary ;  the  spec- 
tator on  the  sun,  who  views  the  motions  of  the  planets  as  they 
actually  exist  in  nature,  would  make  no  such  distinction  between 
the  different  orbits,  but  merely  inquire  how  they  were  mutually 
related  to  each  other.  Taking,  however,  the  ecliptic  as  the  plane 
to  which  all  the  others  are  referred,  we  do  not,  as  in  the  case  of 
the  other  planets,  inquire  how  its  plane  is  inclined,  nor  what  are 
its  nodes,  since  it  has  neither  inclination  nor  node. 

356.  Such,  in  general,  are  the  real  motions  of  the  planets,  and 
such  the  appearances  which  the  planetary  system  would  exhibit 
to  a  spectator  at  the  center  of  motion.     But  in  order  to  represent 
correctly  the  positions  of  the  planetary  orbits,  at  any  given  time, 
three  things  must  be  regarded, — the  Inclination  of  the  orbit  to  the 
ecliptic — the  position  of  the  line  of  the  Nodes— and  the  position  of 
the  line  of  the  Apsides.     In  our  common  diagrams,  the  orbits  are 
incorrectly  represented,  being  all  in  the  same  plane,  as  in  the  fol- 
lowing diagram,  where  AEB  (Fig.  65,)  represents  the  orbit  of 


212 


THE   PLANETS. 

Fig.  65. 


Mercury  as  lying  in  the  same  plane  with  the  ecliptic.  To  exhibit 
its  position  justly  (AB  being  taken  as  the  line  of  the  nodes)  it 
should  be  elevated  on  one  side  about  7°  and  depressed  by  the 
same  number  of  degrees  on  the  other  side,  turning  on  the  line 
AB  as  on  a  hinge.  But  even  then  the  representation  may  be 
incorrect  in  other  respects,  for  we  have  taken  it  for  granted  that 
the  line  of  the  nodes  coincides  with  the  line  of  the  apsides,  or 
that  the  orbit  of  Mercury  cuts  the  ecliptic  in  the  line  AB.  Whereas, 
it  may  lie  in  any  given  position  with  respect  to  the  line  of  the  ap- 
sides depending  on  the  longitude  of  the  nodes.  If,  for  example,  the 
line  of  the  nodes  had  chanced  to  pass  through  Taurus  and  Scor- 
pio instead  of  Cancer  and  Capricorn,  then  it  would  have  been  repre- 
sented by  the  line  8  ^l  instead  of  25V3,  and  the  plane  when  elevated 
or  depressed  with  respect  to  the  plane  of  the  equator,  would  be 
turned  on  this  line  in  our  figure.*  Moreover,  our  diagram  repre- 
sents the  line  of  the  apsides  as  passing  through'Cancer  and  Cap- 
ricorn,  whereas  it  may  have  any  other  position  among  the  signs, 
according  to  the  longitudes  of  the  perigee  and  apogee. 

*  The  learner  will  find  it  useful  to  construct  such  representations  of  the  mutual  re 
lations  of  the  planetary  orbits  of  paste  board. 


MOTIONS  OF  THE  PLANETARY  SYSTEM.  213 

357.  The  attempt  to  exhibit  the  motions  of  the  solar  system,  and 
the  positions  of  the  planetary  orbits  by  means  of  diagrams,  or 
even  orreries,  is  usually  a  failure.     The  student  who  relies  exclu- 
sively on  such  aids  as  these,  will  acquire  ideas  on  this  subject  that 
are  both  inadequate  and  erroneous.     They  may  aid  reflection,  but 
can  never  supply  its  place.     The  impossibility  of  representing 
things  in  their  just  proportions  will  be  evident  when  we  reflect, 
that  to  do  this,  if,  in  an  orrery,  we  make  Mercury  as  large  as  a 
cherry,  we  should  require  to  represent  the  sun  by  a  globe  six  feet 
in  diameter.     If  we  preserve  the  same  proportions  in  regard  to 
distance,  we  must  place  Mercury  250  feet,  and  Uranus  12,500 
feet,  or  more  than  two  miles  from  the  sun.     The  mind  of  the  stu- 
dent of  astronomy  must,  therefore,  raise  itself  from  such  imperfect 
representations  of  celestial  phenomena  as  are  afforded  by  artificial 
mechanism,  and,  transferring  his  contemplations  to  the  celestial 
regions  themselves,  he  must  conceive  of  the  sun  and  planets  as 
bodies  that  bear  an  insignificant  ratio  to  the  immense  spaces  in 
which  they  circulate,  resembling  more  a  few  little  birds  flying  in 
the  open  sky,  than  they  do  the  crowded  machinery  of  an  orrery. 

358.  Having  acquired  as  correct  an  idea  as  we  are  able  of  the 
planetary  system,  and  of  the  positions  of  the  orbits  with  respect  to 
the  ecliptic,  let  us  next  inquire  into  the  nature  and  causes  of  the 
apparent  motions. 

The  apparent  motions  of  the  planets  are  exceedingly  unlike  the 
real  motions,  a  fact  which  is  owing  to  two  causes ;  first,  we  view 
them  out  of  the  center  of  their  orbits  ;  secondly,  we  are  ourselves  in 
motion.  From  the  first  cause,  the  apparent  places  of  the  planets 
are  greatly  changed  by  perspective  ;  and  from  the  second  cause, 
we  attribute  to  the  planets  changes  of  place  which  arise  from  our 
own  motions  of  which  we  are  unconscious. 

359.  The  situation  of  a  heavenly  body  as  seen  from  the  center 
of  the  sun,  is  called  its  heliocentric  place  ;  as  seen  from  the  center 
of  the  earth,  its  geocentric  place.     The  geocentric  motions  of  the 
planets  must,  according  to  what  has  just  been  said,  be  far  more 
irregular  and  complicated  than  the  heliocentric,  as  will  be  evident 
from  the  following  diagram,  which  represents  the  geocentric  mo- 


214 


THE   PLANETS. 


tions  of  Mercury  for  two  entire  revolutions,  embracing  a  period 
of  nearly  six  months. 

Let  S  (Fig.  66,)  represent  the  sun,  1,  2, 3,  &c.  the  orbit  of  Mer- 
cury, a,  b,  c,  &c.  that  of  the  earth,  and  GT  the  concave  sphere  of 
the  heavens.  The  orbit  of  Mercury  is  divided  into  12  equal  parts, 
each  of  which  he  describes  in  7£  days,  and  a  portion  of  the  earth's 

Fig.  66. 


orbit  described  by  that  body  in  the  time  that  Mercury  describes 
the  two  complete  revolutions,  is  divided  into  24  equal  parts.  Let 
us  now  suppose  that  Mercury  is  at  the  point  1  in  his  orbit,  when 
the  earth  is  at  the  point  a  ;  Mercury  will  then  appear  in  the  heav- 
ens at  A.  In  7i  days  Mercury  will  have  reached  2,  while  the 
earth  has  reached  6,  when  Mercury  will  appear  at  B.  By  laying 
a  ruler  on  the  point  c  and  3,  d  and  4,  and  so  on,  in  the  order  of  the 


MOTIONS  OF  THE  PLANETARY  SYSTEM.  215 

alphabet,  the  successive  apparent  places  of  Mercury  in  the  heavens 
will  be  obtained. 

From  A  to  C,  the  apparent  motion  is  direct,  or  in  the  order  of 
the  signs ;  from  C  to  G  it  is  retrograde ;  at  G  it  is  stationary 
awhile,  and  then  direct  through  the  whole  arc  GT.  At  T  the 
planet  is  again  stationary,  and  afterwards  retrograde  along  the 
arc  TX. 

360.  Venus  exhibits  a  variety  of  motions  similar  to  those  of 
Mercury,  except  that  the  changes  do  not  succeed  each  other  so 
rapidly,  since  her  period  of  revolution  approaches  much  more 
nearly  to  that  of  the  earth. 

361.  The  apparent  motions  of  the  superior  planets,  are,  like 
those  of  Mercury  and  Venus,  alternately  direct,  stationary,  and 
retrograde.     In  this  case,  however,  the  earth  moves  faster  than 
the  planet,  and  the  planet  has  its  opposition  but  no  inferior  con- 
junction, whereas  an  inferior  planet  has  its  inferior  conjunction, 
but  no  opposition.     These  differences  render  the  apparent  motions 
of  the  superior  planets  in  some  respects  unlike  those  of  Mercury 
and  Venus.     When  a  superior  planet  is  in  conjunction,  its  motion 
is  direct,  because,  as  in  the  case  of  Venus  in  her  superior  conjunc- 
tion, (see  Fig.  60,)  the  only  effect  of  the  earth's  motion  is  to  ac- 
celerate it;  but   when  the  planet  is  in  opposition,  the  earth   is 
moving  past  it  with  a  greater  velocity,  and  makes  the  planet  seem 
to  move  backwards,  like  the  apparent  backward  motion  of  a  ves- 
sel when  we  overtake  it  and  pass  rapidly  by  it  in  a  steamboat. 

362.  But  the  various  motions  of  a  superior  planet  will  be  best 
understood  from  a  diagram.     Hence,  let  S  (Fig.  67,)  be  the  sun  ; 
B,  C,  D,  E,  the  orbit  of  the  earth ;  &,  c,  d,  &c.  the  orbit  of  a  supe- 
rior planet,  as  Jupiter  for  example  ;  and  I'E'  a  portion  of  the  con- 
cave sphere  of  the  heavens.     Let  bm  be  the  arc  described  by  Ju- 
piter in  the  time  the  earth  describes  the  arc  BM  ;  let  be,  cd,  and 
de,  &c.  be  described  by  Jupiter  while  the  earth  describes  BC, 
CD,  and  DE.     Now  when  the  earth  is  at  B  and  Jupiter  at  bt  he 
will  appear  in  the  heavens  at  B'.     When  the  earth  reaches  C,  the 
planet  reaches  c  and  will  be  seen  at  C',  his  motions  having  been 


218 


direct  from  west  to  east.  While  the  earth  moves  from  C  to  D 
and  from  D  to  E,  Jupiter  has  moved  from  c  to  d,  and  from  d  to 
e,  and  will  appear  to  have  advanced  among  the  stars  from  C'  to 
D',  and  from  D'  to  E',  his  motion  being  still  direct,  but  slower 
than  before,  as  he  has  passed  over  only  the  space  D'E'  in  the 
same  time  that  he  before  moved  through  the  greater  spaces  B'C' 
and  C'D'. 

During  the  motion  of  the  earth  from  E  to  F,  and  of  Jupiter 
from  e  to/,  the  earth  passes  by  Jupiter ;  and  not  being  conscious 
of  our  own  motion,  Jupiter  seems  to  us  to  have  moved  backward 
from  E'  to  F'.  At  E'  where  the  direct  motion  was  changed  to  a 
retrograde,  he  would  appear  to  be  stationary.  Upon  the  arrival 
of  the  earth  at  G,  and  of  Jupiter  at  g,  in  opposition  to  the  sun,  Ju- 
piter will  appear  at  G',  having  moved  with  apparently  great  ve- 
locity over  a  large  space  F'G'.  While  the  earth  passes  from  G  to 
H,  and  from  H  to  I,  and  Jupiter  from  g  to  h,  and  from  h  to  «',  he 


.^TERMINATION  OF  THE  PLANETARY  ORBITS.  217 

will  appear  to  have  moved  from  G'  to  P.  At  P  he  will  again 
appear  stationary  in  the  heavens  ;  but  when  he  advances  from  i  to 
k  in  the  time  the  earth  moves  from  I  to  K,  he  has  described  the 
arch  I'K',  and  has  therefore  resumed  his  direct  motion  from  west 
to  east.  While  the  earth  moves  from  K  to  L  and  from  L  to  M, 
and  Jupiter  through  the  corresponding  spaces  kl  and  Zra,  the  planet 
will  appear  still  to  continue  his  direct  motion  from  K'  to  L'  and 
from  L'  to  M'  in  the  heavens. 

Thus,  during  a  period  of  six  months,  while  the  earth  is  perform- 
ing one  half  of  her  annual  circuit,  Jupiter  has  a  diversity  of  mo- 
tions, all  performed  within  a  small  portion  of  the  heavens. 


CHAPTER    XII. 


DETERMINATION    OF  THE   PLANETARY    ORBITS KEPLER  S    DISCOVER- 
IES  ELEMENTS      OF     THE     ORBIT     OF     A     PLANET QUANTITY     OP 

MATTER    IN    THE    SUN     AND     PLANETS STABILITY    OF    THE    SOLAR 

SYSTEM.* 

363.  IN  chapter  II,  we  have  shown  that  the  figure  of  the  earth's 
orbit  is  an  ellipse,  having  the  sun  in  one  of  its  foci,  and  that  the 
earth's  radius  vector  describes  equal  areas  in  equal  times  ;  and  in 
Chapter  III,  we  have   remarked  that  these  are  only  particular 
examples  under  the  law  of  Universal  Gravitation,  as  is  also  the 
additional  fact,  that  the  squares  of  the  periodical  times  of  the 
planets  are  as  the  cubes  of  the  major  axes  of  their  orbits.     We 
may  now  learn,  more  particularly,  the  process  by  which  the  illus- 
trious Kepler  was  conducted  to  the  discovery  of  these  grand  laws 
of  the  planetary  system. 

364.  Ptolemy,  while  he  held  that  the  orbits  of  the  planets  were 
perfect  circles  in  which  the  planets  revolved  uniformly  about  the 
earth,  was  nevertheless  obliged  to  suppose   that  the  earth  was 
situated  out  of  the  center  of  the  circles,  and  that  at  the  same 

*  See  Article  IV.  of  the  Addenda. 
28 


218  THE   PLANETS. 

distance  on  the  other  side  of  the  center  was  situated  the  point 
(punctum  cequans)  about  which  the  angular  motion  of  the  body 
was  equable  and  uniform.  On  nearly  the  same  suppositions,  Ty- 
cho  Brahe  had  made  a  great  number  of  very  accurate  observations 
on  the  planetary  motions,  which  served  Kepler  as  standards  of 
comparison  for  results,  which  he  deduced  from  calculations  founded 
on  the  application  of  geometrical  reasoning  to  hypotheses  of  his  own. 
Kepler  first  applied  himself  to  investigate  the  orbit  of  Mars,  the 
motions  of  which  planet  appeared  more  irregular  than  those  of 
any  other,  except  Mercury,  which,  being  seldom  seen,  had  then 
been  very  little  studied.  According  to  the  views  of  Ptolemy  and 
Tycho,  he  at  first  supposed  the  orbit  to  be  circular,  and  the  planet 
to  move  uniformly  about  a  point  at  a  certain  distance  from  the 
sun.  He  made  seventy  suppositions  before  he  obtained  one  that 
agreed  with  observation,  the  calculation  of  which  was  extremely 
long  and  tedious,  occupying  him  more  than  five  years.*  The  sup- 
position of  an  equable  motion  in  a  circle,  however  varied,  could 
not  be  made  to  conform  to  the  observations  of  Tycho,  whereas  the 
supposition  that  the  orbit  was  of  an  oval  figure,  depressed  at  the 
sides,  but  coinciding  with  a  circle  at  the  perihelion,  agreed  very 
nearly  with  observation.  Such  a  figure  naturally  suggested  the 
idea  of  an  ellipse,  and  reasoning  on  the  known  properties 
of  the  ellipse,  and  comparing  the  results  of  calculation  with 
actual  observation,  the  agreement  was  such  as  to  leave  no  doubt 
that  the  orbit  of  Mars  is  an  ellipse,  having  the  sun  in  one  of  the 
foci.  He  immediately  conjectured  that  the  same  is  true  of  the 
orbits  of  all  the  other  planets,  and  a  similar  comparison  of  this 
hypothesis  with  observation,  confirmed  its  truth.  Hence  he 
established  the  first  great  law,  that  the  planets  revolve  about  the 
sun  in  ellipses,  having  the  sun  in  one  of  the  foci. 

365.  Kepler  also  discovered  from  observation,  that  the  velocities 
of  the  planets  when  in  their  apsides,  are  inversely  as  their  dis- 
tances from  the  sun,  whence  it  follows  that  they  describe,  in  these 


*  Si  te  hujus  laboriosffl  methodi  pertsesum  fuerit,  jure  mei  te  misereat,  qui  earn  ad 
minimum  septuagies  ivi  cum  plurima  temporis  jactura  ;  et  mirari  desines  hunc  quintum 
Jam  annum  abire,  ex  quo  Martem  aggressus  sum. 


KEPLER  S  DISCOVERIES.  219 

points,  equal  areas  about  the  sun  in  equal  times.  Although  he 
could  not  prove,  from  observation,  that  the  same  was  true  in 
every  point  of  the  orbit,  yet  he  had  no  doubt  that  it  was  so. 
Therefore,  assuming  this  principle  as  true,  and  hence  deducing  the 
equation  of  the  center,  (Art.  200,)  he  found  the  result  to  agree 
with  observation,  and  therefore  concluded  in  general,  that  the 
planets  describe  about  the  sun  equal  areas  in  equal  times. 

366.  Having,  in  his  researches  that  led  to  the  discovery  of  the 
first  of  the  above  laws,  found  the  relative  mean  distances  of  the 
planets  from  the  sun,  and  knowing  their  periodic  times,  Kepler 
next  endeavored  to  ascertain  if  there  was  any  relation  between 
them,  having  a  strong  passion  for  finding  analogies  in  nature. 
He  saw  that  the  more  distant  a  planet  was  from  the  sun,  the 
slower  it  moved ;  so  that  the  periodic  times  of  the  more  distant 
planets  would  be  increased  on  two  accounts,  first,  because  they 
move  over  a  greater  space,  and  secondly,  because  their  motions  in 
their  orbits  are  actually  slower  than  the  motions  of  the  planets 
nearer  the  sun.  Saturn,  for  example,  is  9^  times  further  from  the 
sun  than  the  earth  is,  and  the  circle  described  by  Saturn  is  greater 
than  that  of  the  earth  in  the  same  ratio ;  and  since  the  earth  re- 
volves around  the  sun  in  one  year,  were  their  velocities  equal,  the 
periodic  time  of  Saturn  would  be  9|  years,  whereas  it  is  nearly  30 
years.  Hence  it  was  evident,  that  the  periodic  times  of  the  plan- 
ets increase  in  a  greater  ratio  than  their  distances,  but  in  a  less 
ratio  than  the  squares  of  their  distances,  for  on  that  supposition 
the  periodic  time  of  Saturn  would  be  about  90£  years.  Kepler 
then  took  the  squares  of  the  times  and  compared  them  with  the 
cubes  of  the  distances,  and  found  an  exact  agreement  between 
them.  Thus  he  discovered  the  famous  law,  that  the  squares  of  the 
periodic  times  of  all  the  planets,  are  as  the  cubes  of  their  mean  dis- 
tances from  the  sun.* 

This  law  is  strictly  true  only  in  relation  to  planets  whose  mian- 
thy  of  matter  in  comparison  with  that  of  the  central  body  is 
inappreciable.  When  this  is  not  the  case,  the  periodic  time  is 
shortened  in  the  ratio  of  the  square  root  of  the  sun's  mass  divi- 

*  Vince's  Complete  System,  I,  98. 


220  THE    PLANETS. 

ded  by  the  sun's  plus  the  planet's  mass  /-— )  .     The  mass  of 

\  M+w/ 

most  of  the  planets  is  so  small  compared  with  the  sun's,  that  this 
modification  of  the  law  is  unnecessary  except  where  extreme  ac- 
curacy is  required. 


ELEMENTS  OF  THE  PLANETARY  ORBITS. 

367.  The  particulars  necessary  to  be  known  in  order  to  deter- 
mine the  precise  situation  of  a  planet  at  any  instant,  are  called 
the  Elements  of  its  Orbit.     They  are  seven  in  number,  of  which 
the  first  two  determine  the  absolute  situation  of  the  orbit,  and  the 
other  five  relate  to  the  motion  of  the  planet  in  its  orbit.     These 
elements  are, 

(1.)  The  position  of  the  line  of  the  nodes. 

(2.)  The  inclination  to  the  ecliptic. 

(3.)  The  periodic  time. 

(4.)  The  mean  distance  from  the  sun,  or  semi-axis  major. 

(5.)  The  eccentricity. 

(6.)  The  place  of  the  perihelion. 

(7.)  The  place  of  the  planet  in  its  orbit  at  a  particular  epoch. 

368.  It  may  at  first  view  be  supposed  that  we  can  proceed  to 
find  the  elements  of  the  orbit  of  a  planet  in  the  same  manner  as 
we  did  those  of  the  solar  or  lunar  orbit,  namely,  by  observations 
on  the  right  ascension  and  declination  of  the  body,  converted  into 
latitudes  and  longitudes  by  means  of  spherical  trigonometry,  (See 
Art.  132.)     But  in  the  case  of  the  moon,  we  are  situated  in  the 
center  of  her  motions,  and  the  apparent  coincide  with  the  real 
motions ;  and,  in  respect  to  the  sun,  our  observations  on  his  appa- 
rent motions  give  us  the  earth's  real  motions,  allowing  180°  differ- 
ence in  longitude.     But  as  we  have  already  seen,  the  motions  of 
the  planets  appear  exceedingly  different  to  us,  from  what  they 
would  if  seen  from  the  center  of  their  motions.     It  is  necessary 
therefore  to  deduce  from  observations  made  on  the  earth  the  cor- 
responding results  as  they  would  be  if  viewed  from  the  center  of 
the  sun ;  that  is,  in  the  language  of  astronomers,  having  the  geo- 
centric place  of  a  planet,  it  is  required  to  find  its  heliocentric  place. 


ELEMENTS  OF  THE  PLANETARY  ORBITS.  221 

3G9.  The  first  steps  in  this  process  are  the  same  as  in  the  case  of 
the  sun  and  moon.  That  is,  for  the  purpose  of  finding  the  right 
ascension  and  declination,  the  planet  is  observed  on  the  meridian 
with  the  Transit  Instrument  and  Mural  circle,  (See  Arts.  155  and 
230,)  and  from  these  observations,  the  planet's  geocentric  longitude 
and  latitude  are  computed  by  spherical  trigonometry.  The  distance 
of  the  planet  from  the  sun  is  known  nearly  by  Kepler's  law.  From 
these  data  it  is  required  to  find  the  heliocentric  longitude  and  lati- 
tude. 

Let  S  and  E  (Fig.  68,)  be  the  sun  and  earth,  P  the  planet,  PO 
a  line  drawn  from  P  perpendicular  to  the  ecliptic,  SA  the  direction 
of  Aries,  and  EH  parallel  to  SA,  and  therefore  (on  account  of 
the  immense  distance  of  the  fixed  stars)  also  in  the  direction  of 
Aries.  Then  OEH,  being  the  apparent  distance  of  the  planet 
from  Aries  in  the  direction  of  the  ecliptic,  is  the  geocentric  longi- 
tude, and  OEP,  being  the  apparent  distance  of  the  planet  from  the 
ecliptic  taken  on  a  secondary  to  the  ecliptic,  is  the  geocentric 
latitude.  It  is  obvious  also  that  the  angles  OSA  and  PSO  are 

Fig.  68. 


the  heliocentric  longitude  and  latitude.  The  planet's  angular  dis- 
tance from  the  sun,  PES,  is  also  known  from  observation.  Hence, 
in  the  triangle  SEP,  we  know  SP  and  SE  and  the  angle  SEP,  from 
which  we  can  find  PE ;  and  knowing  PE  and  the  angle  PEO,  we 
can  find  OE,  since  OEP  is  a  right  angled  triangle.  Hence  in  the 
triangle  SEO,  ES  and  EO,  and  the  angle  SEO  (=OEH-SEH= 
difference  of  longitude  of  the  planet  and  the  sun)  are  known,  and 
hence  we  can  obtain  OSE,  (Art.  135,)  which  added  to  the  sun's  lon- 
gitude ESA,  gives  us  OSA  the  planet's  heliocentric  longitude. 


222  THE    PLANETS. 

Also,  because  PS  :  Rad. ::  OP  :  Sin.  PSO 

A  PSxSin.  PSO=OPxRad. 
But  EP  :  Rad. : :  OP  :  Sin.  OEP 

/.  EPxSin.  OEP-OPxRad. 

/.  PSxSin.  PSO=EPxSin.  OEP 

/.  PS  :  EP ::  Sin.  OEP  :  Sin.  PSO. 

The  first  three  terms  of  this  proposition  being  known,  the  last 
is  found,  which  is  the  heliocentric  latitude.* 

370.  Having  now  learned  how  observations  made  at  the  earth 
may  be  converted  into  corresponding  observations  made  at  the 
sun,  we  may  proceed  to  explain  the  mode  of  finding  the  several 
elements  before  enumerated ;  although  our  limits  will  not  permit 
us  to  enter  further  into  detail  on  this  subject,  than  to  explain  the 
leading  principles  on  which  each  of  these  elements  is  determined.-]- 

371.  First,  to  determine  the  position  of  the  Nodes,  and  the  In- 
clination of  the  Orbit. 

These  two  elements,  which  determine  the  situation  of  the  orbit, 
(Art.  367,)  may  be  derived  from  two  heliocentric  longitudes  and 
latitudes.  Let  AR  and  AS  (Fig.  69,)  Fig.  69. 

be  two  heliocentric  longitudes,  PR  and 
QS  the  heliocentric  latitudes,  and  N 
the  ascending  node.  Then,  by  Napier's 

theorem,  (Art.  132,)  Ar~  R    s 

Sin.  NR  (=AR-AN)_  _sin.  NS  (=AS- AN) 

tan.  PR  tan.  QS 

Sin.  ARxcos.  AN  — cos.  ARxsin.  ANJ_ 

tan.  PR. 
sin.  AS  x  cos.  AN— cos.  AS  x sin.  AN 

tan.  QS~ 

sin.  AN_Sin.   ARxtan.   QS— sin.  AS  x  tan.  PR 
But  tan.  AJN    cos.  AN~C^s.  ARxtan.  QS-cos.  ASxtan.  PR* 


*  Brinkley's  Elements  of  Astronomy,  p.  164. 

t  Most  of  these  elements  admit  of  being  determined  in  several  different  ways,  an 
explanation  of  which  may  be  found  in  the  larger  works  on  Astronomy,  as  Vince's  Com- 
plete System,  Vol.  1.  Gregory's  Ast.  p.  212.  Woodhouse,  p.  562. 

t  Day's  Trig.  Art.  208. 


ELEMENTS  OP  THE  PLANETARY  ORBITS.  223 

But  AN  is  the  longitude  of  the  ascending  node ;  and  its  value 
is  found  in  terms  of  the  heliocentric  longitudes  and  latitudes  pre- 
viously determined,  (Art.  369.) 

Again,  since  AN  is  found,  we  may  deduce  from  the  first  equa- 
tion above  the  value  of  PNR,  which  is  the  inclination  of  the  orbit.* 

372.  Secondly,  to  find  the  Periodic  Time. 

This  element  is  learned,  by  marking  the  interval  that  passes 
from  the  time  when  a  planet  is  in  one  of  the  nodes  until  it  returns 
to  the  same  node.  We  may  know  when  a  planet  is  at  the  node 
because  then  its  latitude  is  nothing.  If,  from  a  series  of  observa- 
tions on  the  right  ascension  and  declination  of  a  planet,  we  deduce 
the  latitudes,  and  find  that  one  of  the  observations  gives  the  lati- 
tude 0,  we  infer  that  the  planet  was  at  that  moment  at  the  node. 
But  if,  as  commonly  happens,  no  observation  gives  exactly  0,  then 
we  take  two  latitudes  that  are  nearest  to  0,  but  on  opposite  sides 
of  the  ecliptic,  one  south  and  the  other  north,  and  as  the  sum  of  the 
arcs  of  latitude  is  to  the  whole  interval,  so  is  one  of  the  arcs  to  the 
corresponding  time  in  which  it  was  described,  which  time  being 
added  to  the  first  observation,  or  subtracted  from  the  second,  will 
give  the  precise  moment  when  the  planet  was  at  the  node. 

By  repeated  observations  it  is  found,  that  the  nodes  of  the  planets 
have  a  very  slow  retrograde  motion. 

373.  If  the  orbit  of  a  planet  cut  the  ecliptic  at  right  angles,  then 
small  changes  of  place  would  be  attended  by  appreciable  differ- 
ences of  latitude ;  but  in  fact  the  planetary  orbits  are  in  general 
but  little  inclined  to  the  ecliptic,  and  some  of  them  He  almost  in 
the  same  plane  with  it.     Hence  arises  a  difficulty  in  ascertaining 
the  exact  time  when  a  planet  reaches  its  node.     Among  the  most 
valuable  observations  for  determining  the  elements  of  a  planet's 
orbit,  are  those  made  when  a  superior  planet  is  in  or  near  its  oppo- 
sition to  the  sun,  for  then  the  heliocentric  and  geocentric  longitudes 
are  the  same.     When  a  number  of  oppositions  are  observed,  the 
planet's  motion  in  longitude  as  would  be  observed  from  the  sun  will 
be  known.     The  inferior  planets  also,  when  in  superior  conjunction, 

*  Brinkley,  p.  166. 


224  THE  PLANETS. 

have  their  geocentric  and  heliocentric  longitudes  the  same.  When 
in  inferior  conjunction,  these  longitudes  differ  180° ;  but  the  in- 
ferior planets  can  seldom  be  observed  in  superior  conjunction,  on 
account  of  their  proximity  to  the  sun,  nor  in  inferior  conjunction 
except  in  their  transits,  which  occur  too  rarely  to  admit  of  obser- 
vations sufficiently  numerous.  Therefore,  we  cannot  so  readily 
ascertain  by  simple  observation,  the  motions  of  the  inferior  planets 
seen  from  the  sun,  as  we  can  those  of  the  superior.* 

Hence,  in  order  to  obtain  accurately  the  periodic  time  of  a 
planet,  we  find  the  interval  elapsed  between  two  oppositions  sep- 
arated by  a  long  interval,  when  the  planet  was  nearly  in  the  same 
part  of  the  Zodiac.  From  the  periodic  time,  as  determined  ap- 
proximately by  other  methods,  it  may  be  found  when  the  planet 
has  the  same  heliocentric  longitude  as  at  the  first  observation. 
Hence  the  time  of  a  complete  number  of  revolutions  will  be 
known,  and  thence  the  time  of  one  revolution.  The  greater  the 
interval  of  time  between  the  two  oppositions,  the  more  accurately 
the  periodic  time  will  be  obtained,  because  the  errors  of  observa- 
tion will  be  divided  between  a  great  number  of  periods ;  there- 
fore by  using  very  accurate  observations,  much  precision  may  be 
attained.  For  example,  the  planet  Saturn  was  observed  in  the 
year  228  B.  C.  March  2,  (according  to  our  reckoning  of  time,)  to 
be  near  a  certain  star  called  j  Virginis,  and  it  was  at  the  same 
time  nearly  in  opposition  to  the  sun.  The  same  planet  was  again 
observed  in  opposition  to  the  sun,  and  having  nearly  the  same 
longitude,  in  Feb.  1714.  The  exact  difference  between  these  dates 
was  1943y.  118d.  21h.  15m.  It  is  known  from  other  sources,  that 
the  time  of  a  revolution  is  29%  years  nearly,  and  hence  it  was 
found  that  in  the  above  period  there  were  66  revolutions  of  Saturn ; 
and  dividing  the  interval  by  this  number,  we  obtain  29.444  years, 
which  is  nearly  the  periodic  time  of  Saturn  according  to  the  most 
accurate  determination. 

374.  Thirdly,  to  determine  the  distance  from  the  sun,  and  major 
axes  of  the  planetary  orbits. 

The  distance  of  the  earth  from  the  sun  being  known,  the  mean 

*  Brinkley,  p.  167. 


ELEMENTS  OF  THE  PLANETARY  ORBITS  225 

distance  of  any  planet  (its  periodic  time  being  known)  may  be 
found  by  Kepler's  law,  that  the  squares  of  the  periodic  times  are 
as  the  cubes  of  the  distances.  The  method  of  finding  the  dis- 
tance of  an  inferior  planet  from  the  sun  by  observations  at  the 
greatest  elongation,  has  been  already  explained,  (See  Art.  308.) 
The  distance  of  a  superior  planet  may  be  found  from  observations 
on  its  retrograde  motion  at  the  time  of  opposition.  The  periodic 
times  of  two  planets  being  known,  we  of  course  know  their  mean 
angular  velocities,  which  are  inversely  as  the  times.  Therefore, 
let  Ee  (Fig.  70,)  be  a  very  small  portion  of  the  earth's  orbit,  and 
Mm  a  corresponding  portion  of  that  of  a  superior  planet,  described 
on  the  day  of  opposition,  about  the  sun  S,  on  which  day  the  three 
bodies  lie  in  one  straight  line  SEMX.  Then  the  angle  ESe  and 
MS/w,  representing  the  respective  angular  velocities  of  the  two 


bodies  are  known.  Now  if  em  be  joined,  and  prolonged  to  meet 
SM  continued  in  X,  the  angle  EXe,  which  is  equal  to  the  alternate 
angle  Xey,  being  equal  to  the  retrogradation  of  the  planet  in  the 
same  time  (being  known  from  observation)  is  also  given.  Ee, 
therefore,  and  the  angle  EXe  being  given  in  the  right  angled  tri- 
angle EXe,  the  side  EX  is  easily  calculated,  and  thus  SX  becomes 
known.  Consequently,  in  the  triangle  SwiX,  we  have  given  the 
side  SX,  and  the  two  angles  mSX  and  wXS,  whence  the  other 
sides  Sm  and  wzX  are  easily  determined.  Now  Sm  is  the  radius  of 
the  orbit  of  the  superior  planet  required,  which  in  this  calculation 
is  supposed  circular  as  well  as  that  of  the  earth, — a  supposition  not 
exact,  but  sufficiently  so  to  afford  a  satisfactory  approximation 
to  the  dimensions  of  its  orbit,  and  which,  if  the  process  be  often 
repeated,  in  every  variety  of  situation  at  which  the  opposition  can 
occur,  will  ultimately  afford  an  average  or  mean  value  of  its  dis- 
lance  fully  to  be  depended  on.* 

375.   The  transverse  or  major  axes  of  the  planetary  orbits  remain 

•SirJ.  Herschel. 
29 


226 


THE   PLANETS. 


always  the  same.  Amidst  all  the  perturbations  to  which  other  ele- 
ments of  the  orbit  are  subject,  the  line  of  the  apsides  is  of  the  same 
invariable  length.  It  is  no  matter  in  what  direction  the  planet  may 
be  moving  at  that  moment.  Various  circumstances  will  influence 
the  eccentricity  and  the  position  of  the  ellipse,  but  none  of  them 
affects  its  length. 

376.  Fourthly,  to  determine  the  place  of  the  perihelion — the  epoch 
of  passing  the  perihelion — and  the  eccentricity. 

There  are  various  methods  of  finding  the  eccentricity  of  a 
planet's  orbit  and  the  place  of  the  perihelion,  and  of  course  the 
position  of  the  line  of  the  apsides.  One  is  derived  from  thegreat- 
est  equation  of  the  center,  (Art.  200.)  The  greatest  equation  is  the 
greatest  difference  that  occurs  between  the  mean  and  the  true 
motion  of  a  body  revolving  in  an  ellipse.  It  will  be  necessary 
first  to  explain  the  manner  in  which  the  greatest  equation  is  found. 

Let  AEBF  (Fig.  71,)  be  the  orbit  of  the  planet,  having  the  sun 
in  the  focus  at  S.  In  an  ellipse,  the  square  root  of  the  product  of 
the  semi-axes  gives  the  radius  of  a  circle  of  the  same  area  as  the 


Fig.  71. 


ellipse.*  Therefore  with  the  center 
S,  at  the  distance  SE=VAKxOK, 
describe  the  circle  CEGF,  then  will 
the  area  of  this  circle  be  equal  to  that 
of  the  ellipse.  At  the  same  time  that 
a  planet  departs  from  A  the  aphelion, 
a  body  begins  to  move  with  a  uniform 
motion  from  C  through  the  periphery 
CEGF,  and  performs  a  whole  revolu- 
tion in  the  same  period  that  the  planet 
describes  the  ellipse  ;  the  motion  of 
this  body  will  represent  the  equal  or 
mean  motion  of  the  earth,  and  it  will 
describe  around  S  areas  or  sectors 
of  circles  which  are  proportional  to  the  times,  and  equal  to  the 
elliptic  areas  described  in  the  same  time  by  the  planet.  Let  the 
equal  motion,  or  the  angle  about  S  proportional  to  the  time,  be 


*  Day's  Mensuration. 


ELEMENTS  OF  THE  PLANETARY  ORBITS.  227 

CSM,  and  take  ASP  equal  to  the  sector  CSM ;  then  the  place  of 
the  planet  will  be  P  ;  MSC  will  be  the  mean  anomaly,  (Art.  200,)  . 
DSC  the  true  anomaly,  and  MSD  the  equation  of  the  center.  Since 
the  sectors  CSM  and  ASP  are  equal,  and  the  part  CSD  is  common 
to  both,  PACD  and  SDM  are  equal ;  and  since  the  areas  of  circu- 
lar sectors  are  proportional  to  their  arcs,  the  equation  of  the  center 
is  greatest  when  the  area  ACPD  is  greatest,  that  is,  at  the  point 
E  where  the  ellipse  and  circle  intersect  one  another.  For  when 
the  planet  descends  further,  to  R  for  instance,  the  equation  becomes 
proportional  to  the  difference  of  the  areas  ACE  and  wER,  or  to 
the  area  GBR/w,  V  being  the  situation  of  the  body  moving  equa- 
bly ;  for  the  sector  CSV  will  be  equal  to  the  elliptic  area  ASR, 
and  taking  away  the  common  space  CERS,  then  ACE— RE?«=the 
sector  VSwz=the  equation.  At  the  points  E  and  F,  where  the 
circle  and  ellipse  intersect,  the  radius  vector  of  the  planet  and  the 
radius  of  the  circle  of  equable  motion  are  equal,  and  of  course  those 
radii  then  describe  equal  areas  in  equal  times;  hence,  when  the 
real  motion  of  the  earth  is  equal  to  the  mean  motion,  the  equation 
of  the  center  is  greatest.*'  The  mean  motion  for  any  given  time 
is  easily  found;  for  the  periodic  time  :  360:: the  given  time  :  the 
number  of  degrees  for  that  time.  Observation  shows  when  the 
actual  motion  of  the  planet  is  the  same  with  this. 

377.  Now  the  equation  of  the  center  is  greatest  twice  in  the 
revolution,  on  opposite  sides  of  the  orbit,  as  at  E  and  F,  which 
points  lie  at  equal  distances  from  the  apsides  ;  and  since  the  whole 
arc  EAF  or  EBF  is  known  from  the  time  occupied  in  describing 
it,  therefore,  by  bisecting  this  arc,  we  find  the  points  A  and  B, 
the  aphelion  and  perihelion,  and  consequently  the  position  of  the 
line  of  the  apsides.  The  time  of  describing  the  area  EBF  being 
known,  by  bisecting  this  interval,  we  obtain  the  moment  of  passing 
the  perihelion,  which  gives  us  the  place  of  the  planet  in  its  orbit  at 
a  particular  epoch. 

The  amount  of  the  greatest  equation  obviously  depends  on  the 
eccentricity  of  the  orbit,  since  it  arises  wholly  from  the  departure 
of  the  ellipse  from  the  figure  of  a  perfect  circle ;  hence,  the  greatest 

*  Gregory's  Astronomy,  p.  197. 


228  THE  PLANETS. 

equation  affords  the  means  of  determining  the  eccentricity  itself. 
In  orbits  of  small  eccentricity,  as  is  the  case  with  most  of  the 
planetary  orbits,  it  is  found  that  the  a.rc  which  measures  the  greatest 
equation  is  very  nearly  equal  to  the  distance  between  the  foci,* 
which  always  equals  twice  the  eccentricity,  the  eccentricity  being 
the  distance  from  the  center  to  the  focus.  Consequently,  57°  17' 
44".8f  :  rad. : :  half  the  greatest  equation  :  the  eccentricity. 

The  foregoing  explanations  of  the  methods  of  finding  the  ele- 
ments of  the  orbits,  will  serve  in  general  to  show  the  learner  how 
these  particulars  are  or  may  be  ascertained  ;  yet  the  methods  actu- 
ally employed  are  usually  more  refined  and  intricate  than  these. 
In  astronomy  scarcely  an  element  is  presented  simple  and  unmixed 
with  others.  Its  value  when  first  disengaged,  must  partake  of  the 
uncertainty  to  which  the  other  elements  are  subject ;  and  can  be 
supposed  to  be  settled  to  a  tolerable  degree  of  correctness,  only 
after  multiplied  observations  and  many  revisions.J 

So  arduous  has  been  the  task  of  finding  the  elements  of  the 
planetary  orbits. 

QUANTITY    OF    MATTER   IN    THE    SUN    AND    PLANETS. 

378.  It  would  seem  at  first  view  very  improbable,  that  an  in 
habitant  of  this  earth  would  be  able  to  weigh  the  sun  and  planets, 
and  estimate  the  exact  quantity  of  matter  which  they  severally  con- 
tain.    But  the  principles  of  Universal  Gravitation  conduct  us  to 
this  result,  by  a  process  remarkable  for  its  simplicity.     By  com- 
paring the  relations  of  a  few  elements  that  are  known  to  us,  we 
ascend  to  the  knowledge  of  such  as  appeared  beyond  the  pale  of 
human  investigation.     We  learn  the  quantity  of  matter  in  a  body 
by  the  force  of  gravity  it  exerts.     Let  us  see  how  this  force  is  ascer- 
tained. 

379.  The  quantities  of  matter  in  two  bodies,  may  be  found  in 
terms  of  the  distances  and  periodic  times  of  two  bodies  revolving 
around  them  respectively,  being  as  the  cubes  of  the  distances  divided 
by  the  squares  of  the  periodic  times. 

*  Vince's  Complete  System,  I,  113. 

t  The  value  of  an  arc  equal  to  radius ;  for  3.14159  :  1 : :  180  :  57o  17'  44".8. 

\  Woodhouse,  p.  579. 


QUANTITY    OF   MATTER    IN    THE    SUN    AND    PLANETS.  229 

The  force  of  gravity  G  in  a  body  whose  quantity  of  matter  is 
M  and  distance  D,  varies  directly  as  the  quantity  of  matter,  and 

inversely  as  the  square  of  the  distance  ;  that  is,  G  oc  -p.      But  it 

is  shown  by  writers  on  Central  Forces,  that  the  force  of  gravity 
also  varies  as  the  distance  divided  by  the  square  of  the  periodic 

time,  or  G  a—.      Therefore,  ™ap2,  anc^  M  a-^.     Thus  we  may 

find  the  respective  quantities  of  matter  in  the  earth  and  the  sun, 
by  comparing  the  distance  and  periodic  time  of  the  moon,  revolving 
around  the  earth,  with  the  distance  and  periodic  time  of  the  earth 
revolving  around  the  sun.  For  the  cube  of  the  moon's  distance 
from  the  earth  divided  by  the  square  of  her  periodic  time,  is  to  the 
cube  of  the  earth's  distance  from  the  sun  divided  by  the  square  of 
her  periodic  time,  as  the  quantity  of  matter  in  the  earth  is  to  that 

23S5453     95,000,0003 
m  the  sun.     That  is,  :  :  :  1  :  353,385.  The  most 


exact  determination  of  this  ratio,  gives  for  the  mass  of  the  sun 
354,936  times  that  of  the  earth.  Hence  it  appears  that  the  sun 
contains  more  than  three  hundred  and  fifty-four  thousand  times  as 
much  matter  as  the  earth.  Indeed  the  sun  contains  eight  hundred 
times  as  much  matter  as  all  the  planets. 

Another  view  may  be  taken  of  this  subject  which  leads  to  the 
same  result.  Knowing  the  velocity  of  the  earth  in  its  orbit,  we 
may  calculate  its  centrifugal  force.  Now  this  force  is  counter- 
balanced, and  the  earth  retained  in  its  orbit,  by  the  attraction  of 
the  sun,  which  is  proportional  to  the  quantity  of  matter  in  the  sun. 
Therefore  we  have  only  to  see  what  amount  of  matter  is  required 
in  order  to  balance  the  earth's  centrifugal  force.  It  is  found  that 
the  earth  itself  or  a  body  as  heavy  as  the  earth  acting  at  the  dis- 
tance of  the  sun,  would  be  wholly  incompetent  to  produce  this 
effect,  but  that  in  fact  it  would  take  more  than  three  hundred  and 
fifty-four  thousand  such  bodies  to  do  it. 

380.  The  mass  of  each  of  the  other  planets  that  have  satellites 
may  be  found,  by  comparing  the  periodic  time  of  one  of  its  satel- 
lites with  its  own  periodic  time  around  the  sun.  By  this  means 
we  learn  the  ratio  of  its  quantity  of  matter  to  that  of  the  sun. 


230  THE  PLANETS. 

The  masses  of  those  planets  which  have  no  satellites,  as  Venus  or 
Mars,  have  been  determined,  by  estimating  the  force  of  attraction 
which  they  exert  in  disturbing  the  motions  of  other  bodies.  Thus, 
the  effect  of  the  moon  in  raising  the  tides,  leads  to  a  knowledge 
of  the  quantity  of  matter  in  the  moon  ;  and  the  effect  of  Venus  in 
disturbing  the  motions  of  the  earth,  indicates  her  quantity  of  mat- 
ter.* 

381.  The  quantity  of  matter  in  bodies  varies  as  their  magnitudes 
and  densities  conjointly.  Hence,  their  densities  vary  as  their 
masses  divided  by  their  magnitudes ;  and  since  we  know  the  mag- 
nitudes of  the  planets,  and  can  compute  as  above  their  masses,  we 
can  thus  learn  their  densities,  which,  when  reduced  to  a  common 
standard,  give  us  their  specific  gravities,  or  show  us  how  much 
heavier  they  are  than  water.  Worlds  therefore  are  weighed  with 
almost  as  much  ease  as  a  pebble,  or  an  article  of  merchandize. 

The  densities  and  specific  gravities  of  the  sun,  moon,  and  planets, 
are  estimated  as  follows  :f 

Density.  Specific  Gravity. 

Sun,          ....  0.2543  1.40J 

Moon,       ....  0.6150  3.37 

Mercury,  ....  2.7820  15.24 

Venus,       ....  0.9434  5.17 

Earth,       .         .         .  1.0000  5.48 

Mars,        .  0.1293  0.71 

Jupiter,     ....  0.2589  1.42 

Saturn,     ....  0.1016  0.56 

Uranus 0.2797  1.53 

From  this  table  it  appears,  that  the  sun  consists  of  matter  but 
little  heavier  than  water ;  but  that  the  moon  is  more  than  three 
times  as  heavy  as  water,  though  less  dense  than  the  earth.  It  also 
appears  that  the  planets  near  the  sun  are,  as  a  general  fact,  more 

*  These  estimates  are  made  by  the  most  profound  investigations  in  La  Place's  Me. 
canique  Celeste,  Vol.  III. 

t  Francoeur. 

{  The  earth  being  taken,  according  to  Bailly,  at  5.48,  the  specific  gravities  of  the 
other  bodies  (which  are  found  by  multiplying  the  density  of  each  by  the  specific  gravity 
of  the  earth)  are  here  stated  somewhat  higher  than  they  are  given  in  most  works. 


STABILITY  OF  THE  SOLAR   SYSTEM.  231 

dense  than  those  more  remote,  Mercury  being  as  heavy  as  the 
heaviest  metals  except  two  or  three,  while  Saturn  is  as  light  as  a 
cork.  The  decrease  of  density  however  is  not  entirely  regular, 
since  Venus  is  a  little  lighter  than  the  earth,  while  Jupiter  is 
heavier  than  Mars,  and  Uranus  than  Saturn. 

382.  The  perturbations  occasioned  in  the  motions  of  the  planets 
by  their  action  on  each  other  are  very  numerous,  since  every  body 
in  the  system  exerts  an  attraction  on  every  other,  in  conformity 
with  the  law  of  Univ  ersal  Gravitation.  Venus  and  Mars,  approach- 
ing as  they  do  at  times  comparatively  near  to  the  earth,  sensibly 
disturb  its  motions,  and  the  satellites  of  the  remote  planets  greatly 
disturb  each  other's  movements. 


STABILITY  OF  THE  SOLAR   SYSTEM. 

383.  The  derangement  which  the  planets  produce  in  the  motion 
of  one  of  their  number  will  be  very  small  in  the  course  of  one 
revolution  ;  but  this  gives  us  no  security  that  the  derangement  may 
not  become  very  large  in  the  course  of  many  revolutions.  The 
cause  acts  perpetually,  and  it  has  the  whole  extent  of  time  to  work 
in.  Is  it  not  easily  conceivable  then  that  in  the  lapse  of  ages,  the 
derangements  of  the  motions  of  the  planets  may  accumulate,  the 
orbits  may  change  their  form,  and  their  mutual  distances  may  be 
much  increased  or  diminished?  Is  it  not  possible  that  these 
changes  may  go  on  without  limit,  and  end  in  the  complete  subver- 
sion and  ruin  of  the  system  ?  If,  for  instance,  the  result  of  this 
mutual  gravitation  should  be  to  increase  considerably  the  eccen- 
tricity of  the  earth's  orbit,  or  to  make  the  moon  approach  contin- 
ally  nearer  and  nearer  to  the  earth  at  every  revolution,  it  is  easy 
to  see  that  in  the  one  case,  our  year  would  change  its  character, 
producing  a  far  greater  irregularity  in  the  distribution  of  the  solar 
heat :  in  the  other,  our  satellite  must  fall  to  the  earth,  occasioning 
a  dreadful  catastrophe.  If  the  positions  of  the  planetary  orbits 
with  respect  to  that  of  the  earth,  were  to  change  much,  the  plan- 
ets might  sometimes  come  very  near  us,  and  thus  increase  the 
effect  of  their  attraction  beyond  calculable  limits.  Under  such 
circumstances  we  might  have  years  of  unequal  length,  and  seasons 


THE    PLANETS. 


of  capricious  temperature  ;  planets  and  moons  of  portentous  size 
and  aspect  glaring  and  disappearing  at  uncertain  intervals ;  tides 
like  deluges  sweeping  over  whole  continents ;  and,  perhaps,  the 
collision  of  two  of  the  planets,  and  the  consequent  destruction  of 
all  organization  on  both  of  them.  The  fact  really  is,  that  changes 
are  taking  place  in  the  motions  of  the  heavenly  bodies,  which  have 
gone  on  progressively  from  the  first  dawn  of  science.  The  eccen- 
tricity of  the  earth's  orbit  has  been  diminishing  from  the  earliest 
observations  to  our  times.  The  moon  has  been  moving  quicker 
from  the  time  of  the  first  recorded  eclipses,  and  is  now  in  advance 
by  about  four  times  her  own  breadth,  of  what  her  own  place 
would  have  been  if  it  had  not  been  affected  by  this  acceleration. 
The  obliquity  of  the  ecliptic  also,  is  in  a  state  of  diminution,  and  is 
now  about  two  fifths  of  a  degree  less  than  it  was  in  the  time  of 
Aristotle.* 

384.  But  amid  so  many  seeming  causes  of  irregularity,  and  ruin, 
it  is  worthy  of  grateful  notice,  that  effectual  provision  is  made  for 
the  stability  of  the  solar  system.  The  full  confirmation  of  this  fact^ 
is  among  the  grand  results  of  Physical  Astronomy.  Newton  did 
not  undertake  to  demonstrate  either  the  stability  or  instability  of 
the  system.  The  decision  of  this  point  required  a  great  number 
of  preparatory  steps  and  simplifications,  and  such  progress  in  the 
invention  and  improvement  of  mathematical  methods  as  occu- 
pied the  best  mathematicians  of  Europe  for  the  greater  part  of 
the  last  century.  Towards  the  end  of  that  time,  it  was  shown  by 
La  Grange  and  La  Place,  that  the  arrangements  of  the  solar  sys- 
tem are  stable  ;  that,  in  the  long  run,  the  orbits  and  motions  remain 
unchanged  ;  and  that  the  changes  in  the  orbits,  which  take  place 
in  shorter  periods,  never  transgress  certain  very  moderate  limits. 
Each  orbit  undergoes  deviations  on  this  side  and  on  that  side  of  its 
average  state  ;  but  these  deviations  are  never  very  great,  and  it 
finally  recovers  from  them,  so  that  the  average  is  preserved.  The 
planets  produce  perpetual  perturbations  in  each  other's  motions, 
but  these  perturbations  are  not  indefinitely  progressive,  but  period- 
ical, reaching  a  maximum  value  and  then  diminishing.  The  pe- 

*  Whewell,  in  the  Bridgewater  Treatises,  p.  128. 


STABILITY  OF  THE  SOLAR   SYSTEM.  233 

riods  which  this  restoration  requires  are  for  the  most  part  enor- 
mous— not  less  than  thousands,  and  in  some  instances  millions  of 
years.  Indeed  some  of  these  apparent  derangements,  have  been 
going  on  in  the  same  direction  from  the  creation  of  the  world. 
But  the  restoration  is  in  the  sequel  as  complete  as  the  derange- 
ment ;  and  in  the  mean  time  the  disturbance  never  attains  a  suf- 
ficient amount  seriously  to  affect  the  stability  of  the  system.*  I 
have  succeeded  in  demonstrating  (says  La  Place)  that,  whatever  be 
the  masses  of  the  planets,  in  consequence  of  the  fact  that  they  all 
move  in  the  same  direction,  in  orbits  of  small  eccentricity,  and  but 
slightly  inclined  to  each  other,  their  secular  irregularities  are  pe- 
riodical and  included  within  narrow  limits ;  so  that  the  planetary 
system  will  only  oscillate  about  a  mean  state,  and  will  never  de- 
viate from  it  except  by  a  very  small  quantity.  The  ellipses  of  the 
planets  have  been  and  always  will  be  nearly  circular.  The  eclip- 
tic will  never  coincide  with  the  equator  ;  and  the  entire  extent  of 
the  variation  in  its  inclination,  cannot  exceed  three  degrees. 

385.  To  these  observations  of  La  Place,  Professor  Whewellf 
adds  the  following  on  the  importance,  to  the  stability  of  the  solar 
system,  of  the  fact  that  those  planets  which  have  great  masses 
have  orbits  of  small  eccentricity.  The  planets  Mercury  and  Mars, 
which  have  much  the  largest  eccentricity  among  the  old  planets, 
are  those  of  which  the  masses  are  much  the  smallest.  The  mass 
of  Jupiter  is  more  than  two  thousand  times  that  of  either  of  these 
planets.  If  the  orbit  of  Jupiter  were  as  eccentric  as  that  of  Mer- 
cury, all  the  security  for  the  stability  of  the  system,  which  analy- 
sis has  yet  pointed  out,  would  disappear.  The  earth  and  the 
smaller  planets  might  in  that  case  change  their  nearly  circular  or- 
bits into  very  long  ellipses,  and  thus  might  fall  into  the  sun,  or  fly 
off  into  remote  space.  It  is  further  remarkable  that  in  the  newly 
discovered  planets,  of  which  the  orbits  are  still  more  eccentric 
than  that  of  Mercury,  the  masses  are  still  smaller,  so  that  the  same 
provision  is  established  in  this  case  also. 

*  Whewell,  in  the  Bridgewater  Treatises,  p.  128. 
t  Bridgewater  Treatises,  p.  131.    See  also  Playfair's  Outlines,  2,  290. 

30 


CHAPTER  XIII. 

OF    COMETS. 

386.  A  COMET,*  when  perfectly  formed,  consists  of  three  parts, 
the  Nucleus,  the  Envelope,  and  the  Tail.     The  Nucleus,  or  body 
of  the  comet,  is  generally  distinguished  by  its  forming  a  bright 
point  in  the  center  of  the  head,  conveying  the  idea  of  a  solid,  or  at 
least  of  a  very  dense  portion  of  matter.     Though  it  is  usually 
exceedingly  small  when  compared  with  the  other  parts  of  the 
comet,  yet  it  sometimes  subtends  an  angle  capable  of  being  meas- 
ured by  the  telescope.     The  Envelope,  (sometimes  called  the  coma,) 
is  a  dense  nebulous  covering,  which  frequently  renders  the  edge 
of  the  nucleus  so  indistinct,  that  it  is  extremely  difficult  to  ascer- 
tain its  diameter  with  any  degree  of  precision.     Many  comets  have 
no  nucleus,  but  present  only  a  nebulous  mass  extremely  attenuated 
on  the  confines,  but  gradually  increasing  in  density  towards  the 
center.     Indeed  there  is  a  regular  gradation  of  comets,  from  such 
as  are  composed  merely  of  a  gaseous  or  vapory  medium,  to  those 
which  have  a  well  defined  nucleus.     In  some  instances  on  record, 
astronomers  have- detected  with  their  telescopes  small  stars  through 
the  densest  part  of  a  comet. 

The  Tail  is  regarded  as  an  expansion  or  prolongation  of  the 
coma ;  and,  presenting  as  it  sometimes  does,  a  train  of  appalling 
magnitude,  and  of  a  pale,  portentous  light,  it  confers  on  this  class 
of  bodies  their  peculiar  celebrity. 

387.  The  number  of  comets  belonging  to  the  solar  system,  is 
probably  very  great.     Many,  no  doubt,  escape  observation  by  being 
above  the  horizon  in  the  day  time.     Seneca  mentions,  that  during 
a  total  eclipse  of  the  sun,  which  happened  60  years  before  the 
Christian  era,  a  large   and  splendid  comet   suddenly   made   its 

t  K6{ui,  coma,  from  the  bearded  appearance  of  comets. 


COMETS. 


235 


Fig.  71'. 


Fig.  71". 


COMET   OF  1811.  COMET  OF  1680. 

appearance,  being  very  near  the  sun.  The  elements  of  at  least  1 30 
have  been  computed,  and  arranged  in  a  table  for  future  compari- 
son. Of  these  six  are  particularly  remarkable,  viz.  the  comets  of 
1680,  1770,  and  1811  ;  and  those  which  bear  the  names  of  Halley, 
Biela,  and  Encke.  The  cornet  of  1680,  was  remarkable  not  only 
for  its  astonishing  size  and  splendor,  and  its  near  approach  to  the 
sun,  but  is  celebrated  for  having  submitted  itself  to  the  observa- 
tions of  Sir  Isaac  Newton,  and  for  having  enjoyed  the  signal  honor 
of  being  the  first  comet  whose  elements  were  determined  on  the 
sure  basis  of  mathematics.  The  comet  of  1770,  is  memorable  for 
the  changes  its  orbit  has  undergone  by  the  action  of  Jupiter,  as 
will  be  more  particularly  related  in  the  sequel.  The  comet  of 
1811  was  the  most  remarkable  in  its  appearance  of  all  that  have 
been  seen  in  the  present  century.  Halley's  comet  (the  same 
which  re-appeared  in  1835)  is  distinguished  as  that  whose  return 
was  first  successfully  predicted,  and  whose  orbit  was  first  deter- 
mined ;  and  Biela's  and  Encke's  comets  are  well  known,  for  their 


236  COMETS. 

short  periods  of  revolution,  which  subject  them  frequently  to  the 
view  of  astronomers. 

388.  In  magnitude  and  brightness  comets  exhibit  a  great  diver- 
sity. History  informs  us  of  comets  so  bright  as  to  be  distinctly 
visible  in  the  day  time,  even  at  noon  and  in  the  brightest  sunshine. 
Such  was  the  comet  seen  at  Rome  a  little  before  the  assassination 
of  Julius  Caesar.  The  comet  of  1680  covered  an  arc  of  the 
heavens  of  97°,  and  its  length  was  estimated  at  123,000,000 
miles.*  That  of  1811,  had  a  nucleus  of  only  428  miles  in  diame- 
ter, but  a  tail  132,000,000  miles  long.f  Had  it  been  coiled  around 
the  earth  like  a  serpent,  it  would  have  reached  round  more  than 
5,000  times.  Other  comets  are  of  exceedingly  small  dimensions, 
the  nucleus  being  estimated  at  only  25  miles  ;  and  some  which  are 
destitute  of  any  perceptible  nucleus,  appear  to  the  largest  tele- 
scopes, even  when  nearest  to  us,  only  as  a  small  speck  of  fog,  or 
as  a  tuft  of  down.  The  majority  of  comets  can  be  seen  only  by 
the  aid  of  the  telescope. 

The  same  comet,  indeed,  has  often  very  different  aspects,  at  its 
different  returns.  Halley's  comet  in  1305  was  described  by  the 
historians  of  that  age,  as  cometa  horrendce  magnitudinis ;  in  1456 
its  tail  reached  from  the  horizon  to  the  zenith,  and  inspired  such 
terror,  that  by  a  decree  of  the  Pope  of  Rome,  public  prayers  were 
offered  up  at  noon-day  in  all  the  Catholic  churches  to  deprecate 
the  wrath  of  heaven,  while  in  1682,  its  tail  was  only  30°  in  length, 
and  in  1759  it  was  visible  only  to  the  telescope,  until  after  it  had 
passed  its  perihelion.  At  its  recent  return  in  1835,  the  greatest 
length  of  the  tail  was  about  12°.J  These  changes  in  the  appear- 
ances of  the  same  comet  are  partly  owing  to  the  different  positions 
of  the  earth  with  respect  to  them,  being  sometimes  much  nearer 
to  them  when  they  cross  its  track  than  at  others ;  also  one  specta 
tor  so  situated  as  to  see  the  comet  at  a  higher  angle  of  elevation  or 
in  a  purer  sky  than  another,  will  see  the  train  longer  than  it 
appears  to  another  less  favorbly  situated ;  but  the  extent  of  the 


*  Arago.  t  Milne's  Prize  Essay  on  Comets, 

t  But  might  be  seen  much  longer  by  indirect  vision.    (Pro/.  Joslin,  Am.  Jour.  Sci- 
ence, 31,  328.) 


COMETS.  237 

changes  are  such  as  indicate  also  a  real  change  in  their  magnitude 
and  brightness. 

389.  The  periods  of  comets  in  their  revolutions  around  the  sun, 
are  equally  various.     Encke's  comet,  which  has  the  shortest  known 
period,  completes  its  revolution  in  3£  years,  or  more  accurately, 
in  1208  days  ;  while  that  of  1811  is  estimated  to  have  a  period  of 
3383  years.* 

390.  The  distances  to  which  different  comets  recede  from  the  sun, 
are  also  very  various.     While  Encke's  comet  performs  its  entire 
revolution  within  the  orbit  of  Jupiter,  Halley's  comet  recedes  from 
the  sun  to  twice  the  distance  of  Uranus,  or  nearly  3600,000,000 
miles.     Some  comets,  indeed,  are  thought  to  go  to  a  much  greater 
distance  from  the  sun  than  this,  while  some  even  are  supposed  to 
pass  into  parabolic  or  hyperbolic  orbits,  and  never  to  return. 

391.  Comets  shine  by  reflecting  the  light  of  the  sun.     In  one  or 
two  instances  they  have  exhibited  distinct  phases,^  although  the 
nebulous  matter  with  which  the  nucleus  is  surrounded,  would  com- 
monly prevent  such  phases  from  being   distinctly  visible,  even 
when   they  would   otherwise  be   apparent.     Moreover,   certain 
qualities  of  polarized  light  enable  the  optician  to  decide  whether 
the  light  of  a  given  body  is  direct  or  reflected ;  and  M.  Arago,  of 
Paris,  by  experiments  of  this  kind  on  the  light  of  the  comet  of 
1819,  ascertained  it  to  be  reflected  light.  J 

392.  The  tail  of  a  comet  usually  increases  veiy  much  as  it 
approaches  the  sun ;  and  frequently  does  not  reach  its  maximum 
until  after  the  perihelion  passage.     In  receding  from  the  sun,  the 
tail  again  contracts,  and  nearly  or  quite  disappears  before  the  body 
of  the  comet  is  entirely  out  of  sight.     The  tail  is  frequently  divi- 
ded into  two  portions,  the  central  parts,  in  the  direction  of  the 
axis,  being1  less  bright  than  the  marginal  parts.     In  1744,  a  comet 
appeared  which  had  six  tails,  spread  out  like  a  fan. 

The  tails  of  comets  extend  in  a  direct  line  from  the  sun,  although 
they  are  usually  more  or  less  curved,  like  a  long  quill  or  feather, 

*  Milne.  t  Delambre,  t.  3,  p  400.  t  Francceur,  181. 


238  COMETS. 

being  convex  on  the  side  next  to  the  direction  in  which  they 
are  moving ;  a  figure  which  may  result  from  the  less  velocity  of 
the  portions  most  remote  from  the  sun.  Expansions  of  the  Enve- 
lope have  also  been  at  times  observed  on  the  side  next  the  sun,* 
but  these  seldom  attain  any  considerable  length. 

393.  The  quantity  of  matter  in  comets  is   exceedingly  small. 
Their  tails  consist  of  matter  of  such  tenuity  that  the  smallest  stars 
are  visible  through  them.     They  can  only  be  regarded  as  great 
masses  of  thin  vapor,  susceptible  of  being  penetrated  through 
their  whole  substance  by  the  sunbeams,  and  reflecting  them  alike 
from  their  interior  parts  and  from  their  surfaces.     It  appears,  per- 
haps, incredible  that  so  thin  a  substance  should  be  visible  by  re- 
flected light,  and  some  astronomers  have  held  that  the  matter  of 
comets  is  self-luminous ;  but  it  requires  but  very  little  light  to  ren- 
der an  object  visible  in  the  night,  and  a  light  vapor  may  be  visible 
when  illuminated  throughout  an  immense  stratum,  which  could  not 
,be  seen  if  spread  over  the  face  of  the  sky  like  a  thin  cloud.     The 
highest  clouds  that  float  in  our  atmosphere,  must  be  looked  upon 
as  dense  and  massive  bodies,  compared  with  the  filmy  and  all  but 
spiritual  texture  of  a  comet.f 

394.  The  small  quantity  of  matter  in  comets  is  proved  by  the 
fact  that  they  have  sometimes  passed  very  near  to  some  of  the  planets 
without  disturbing  their  motions  in  any  appreciable  degree.     Thus 
the  comet  of  1770,  in  its  way  to  the  sun,  got  entangled  among  the 
satellites  of  Jupiter,  and  remained  near  them  four  months,  yet  it  did 
not  perceptibly  change  their  motions.     The  same  comet  also  came 
very  near  the  earth ;  so  near,  that,  had  its  mass  been  equal  to  that 
of  the  earth,  it  would  have  caused  the  earth  to  revolve  in  an  orbit 
so  much  larger  than  at  present,  as  to  have  increased  the  length  of 
the  year  2h.  47m. J     Yet  it  produced  no  sensible  effect  on  the 
length  of  the  year,  and  therefore  its  mass,  as  is  shown  by  La 
Place,  could  not  have  exceeded  -oVo-  °f  tnat  °f  tne  earth,  -  and 
might  have  been  less  than  this  to  any  extent.     It  may  indeed  be 


*  See  Dr.  Joslin's  remarks  on  Halley's  comet,  Amer.  Jour.  Science,  Vol.  31. 
t  Sir.  J.  Herschel.  \  La  Place. 


COMETS.  239 

asked,  what  proof  we  have  that  comets  have  any  matter,  and  are  not 
mere  reflexions  of  light.  The  answer  is  that,  although  they  are  not 
able  by  their  own  force  of  attraction  to  disturb  the  motions  of  the 
planets,  yet  they  are  themselves  exceedingly  disturbed  by  the  action 
of  the  planets,  and  in  exact  conformity  with  the  laws  of  universal 
gravitation.  A  delicate  compass  may  be  greatly  agitated  by  the 
vicinity  of  a  mass  of  iron,  while  the  iron  is  not  sensibly  affected 
by  the  attraction  of  the  needle. 

By  approaching  very  near  to  a  large  planet,  a  comet  may  have 
its  orbit  entirely  changed.  This  fact  is  strikingly  exemplified  in 
the  history  of  the  comet  of  1770.  At  its  appearance  in  1770, 
its  orbit  was  found  to  be  an  ellipse,  requiring  for  a  complete  revo- 
lution only  5|  years ;  and  the  wonder  was,  that  it  had  not  been 
seen  before,  since  it  was  a  very  large  and  bright  comet.  Astron- 
omers suspected  that  its  path  had  been  changed,  and  that  it  had 
been  recently  compelled  to  move  in  this  short  ellipse,  by  the  dis- 
turbing force  of  Jupiter  and  his  satellites.  The  French  Institute, 
therefore,  offered  a  high  prize  for  the  most  complete  investigation 
of  the  elements  of  this  comet,  taking  into  account  any  circum- 
stances which  could  possibly  have  produced  an  alteration  in  its 
course.  By  tracing  back  the  movements  of  this  comet  for  some 
years  previous,  to  1770,  it  was  found  that,  at  the  beginning  of 
1767,  it  had  entered  considerably  within  the  sphere  of  Jupiter's 
attraction.  Calculating  the  amount  of  this  attraction  from  the 
known  proximity  of  the  two  bodies,  it  was  found  what  must  have 
been  its  orbit  previous  to  the  time  when  it  became  subject  to  the 
disturbing  action  of  Jupiter.  The  result  showed  that  it  then 
moved  in  an  ellipse  of  greater  extent,  having  a  period  of  50  years, 
and  having  its  perihelion  instead  of  its  aphelion  near  Jupiter.  It 
was  therefore  evident  why,  as  long  as  it  continued  to  circulate  in 
an  orbit  so  far  from  the  center  of  the  system,  it  was  never  visible 
from  the  earth.  In  January,  1767,  Jupiter  and  the  comet  happened 
to  be  very  near  one  another,  and  as  both  were  moving  in  the  same 
direction,  and  nearly  in  the  same  plane,  they  remained  in  the 
neighborhood  of  each  other  for  several  months,  the  planet  being 
between  the  comet  and  the  sun.  The  consequence  was,  that  the 
comet's  orbit  was  changed  into  a  smaller  ellipse,  in  which  its  revo- 
lution was  accomplished  in  5£  years.  But  as  it  was  approaching 


240  COMETS. 

the  sun  in  1779,  it  happened  again  to  fall  in  with  Jupiter.  It  was 
in  the  month  of  June,  that  the  attraction  of  the  planet  began  to 
have  a  sensible  effect ;  and  it  was  not  until  the  month  of  October 
following  that  they  were  finally  separated. 

At  the  time  of  their  nearest  approach,  in  August,  Jupiter  was 
distant  from  the  comet  only  T^T  of  its  distance  from  the  sun,  and 
exerted  an  attraction  upon  it  225  times  greater  than  that  of  the 
sun.  By  reason  of  this  powerful  attraction,  Jupiter  being  further 
from  the  sun  than  the  comet,  the  latter  was  drawn  out  into  a  new 
orbit,  which  even  at  its  perihelion  came  no  nearer  to  the  sun  than 
the  planet  Ceres.  In  this  third  orbit,  the  comet  requires  about  20 
years  to  accomplish  its  revolution;  and  being  at  so  great  a  dis- 
tance from  the  earth,  it  is  invisible,  and  will  forever  remain  so  un- 
less, in  the  course  of  ages,  it  may  undergo  new  perturbations,  and 
move  again  in  some  smaller  orbit  as  before.* 


ORBITS    AND  MOTIONS    OF    COMETS. 

395.  The  planets,  as  we  have  seen,  (with  the  exception  of  the 
four  new  ones,  which  seem  to  be  an  intermediate  class  of  bodies 
between  planets  and  comets,)  move  in  orbits  which  are  nearly  cir- 
cular, and  all  very  near  to  the  plane  of  the  ecliptic,  and  all  move 
in  the  same  direction  from  west  to  east.     But  the  orbits  of  comets 
are  far  more  eccentric  than  those  of  the  planets ;  they  are  in- 
clined to  the   ecliptic  at  various  angles,  being  sometimes  even 
nearly  perpendicular  to  it ;  and  the  motions  of  comets  are  some- 
times retrograde. 

396.  The  Elements  of  a  comet  are  five,  viz.  (1)  The  perihelion 
distance  ;  (2)  longitude  of  the  perihelion  ;  (3)  longitude  of  the  node  ; 
(4)  inclination  of  the  orbit  ;  (5)  time  of  the  perihelion  passage. 

The  investigation  of  these  elements  is  a  problem  extremely  in- 
tricate, requiring  for  its  solution,  a  skilful  and  laborious  applica- 
tion of  the  most  refined  analysis.  Newton  himself,  pronounced  it 
Problema  longe  difficilimum ;  and  with  all  the  advantages  of  the 
most  improved  state  of  science,  the  determination  of  a  comet's 

*  Milne. 


ORBITS    AND    MOTIONS    OP    COMETS.  241 

orbit  is  considered  one  of  the  most  complicated  problems  in  as- 
tronomy. This  difficulty  arises  from  several  circumstances  pecu- 
liar to  comets.  In  the  first  place,  from  the  elongated  form  of  the 
orbits  which  these  bodies  describe,  it  is  during  only  a  very  small 
portion  of  their  course,  that  they  are  visible  from  the  earth,  and 
the  observations  made  in  that  short  period,  cannot  afterwards  be 
verified  on  more  convenient  occasions  ;  whereas  in  the  case  of  the 
planets,  whose  orbits  are  nearly  circular,  and  whose  movements  may 
be  followed  uninterruptedly  throughout  a  complete  revolution,  no 
such  impediments  to  the  determination  of  their  orbits  occur.  There 
is  also  some  unavoidable  uncertainty  in  observations  made  upon 
bodies  whose  outlines  are  so  ill-defined.  In  the  second  place,  there 
are  many  comets  which  move  in  a  direction  opposite  to  the  order 
of  the  signs  in  the  zodiac,  and  sometimes  nearly  perpendicular  to 
the  plane  of  the  ecliptic  ;  so  that  their  apparent  course  through  the 
heavens  is  rendered  extremely  complicated,  on  account  of  the  con- 
trary motion  of  .he  earth.  In  the  third  place,  as  there  may  be 
a  multitude  of  elliptic  orbits,  whose  perihelion  distances  are  equal, 
it  is  obvious  that,  in  the  case  of  very  eccentric  orbits,  the  slightest 
change  in  the  position  of  the  curve  near  the  vertex,  where  alone 
the  comet  can  be  observed,  must  occasion  a  very  sensible  differ- 
ence in  the  length  of  the  orbit  (as  will  be  obvious  from  Fig.  II"1 ;) 
and  therefore,  though  a  small  error  produces  no  perceptible  dis- 
crepancy between  the  observed  and  the  calculated  course,  while 

Fig.  71"'. 


242  COMETS. 

the  comet  remains  visible  from  the  earth,  its  effect  when  diffused 
over  the  whole  extent  of  the  orbit,  may  acquire  a  most  material  or 
even  a  fatal  importance. 

On  account  of  these  circumstances,  it  is  found  exceedingly  diffi- 
cult to  lay  down  the  path  which  a  comet  actually  follows  through 
the  whole  system,  and  least  of  all,  possible  to  ascertain  with  accu- 
racy, the  length  of  the  major  axis  of  the  ellipse,  and  consequently 
the  periodical  revolution.*  An  error  of  only  a  few  seconds  may 
cause  a  difference  of  many  hundred  years.  In  this  manner,  though 
Bessel  determined  the  revolution  of  the  comet  of  17G9  to  be  2089 
years,  it  was  found  that  an  error  of  no  more  than  5"  in  observation, 
would  alter  the  period  either  to  2678  years,  or  to  1692  years. 
Some  astronomers,  in  calculating  the  orbit  of  the  great  comet  of 
1680,  have  found  the  length  of  its  greater  axis  426  times  the 
earth's  distance  from  the  sun,  and  consequently  its  period  8792 
years  ;  whilst  others  estimate  the  greater  axis  430  times  the  earth's 
distance,  which  alters  the  period  to  8916  years.  Newton  and 
Halley,  however,  judged  that  this  comet  accomplished  its  revolu- 
tion in  only  570  years. 

397.  Disheartened  by  the  difficulty  of  attaining  to  any  precision 
in  that  circumstance,  by  which  an  elliptic  orbit  is  characterized, 
and,  moreover,  taking  into  account  the  laborious  calculations 
necessary  for  ifs  investigation,  astronomers  usually  satisfy  them- 
selves with  ascertaining  the  elements  of  a  comet  on  the  supposition 
of  its  describing  a  parabola ;  and,  as  this  is  a  curve  whose  axis  is 
infinite,  the  procedure  is  greatly  simplified  by  leaving  entirely  out 
of  consideration  the  periodic  revolution.  It  is  true  that  a  parabola 
may  not  represent  with  mathematical  strictness  the  course  which 
a  comet  actually  follows ;  but  as  a  parabola  is  the  intermediate 
curve  between  the  hyperbola  and  ellipse,  it  is  found  that  this 
method,  which  is  so  much  more  convenient  for  computation,  also 
accords  sufficiently  with  observations,  except  in  cases  when  the 
ellipse  is  a  comparatively  short  one,  as  that  of  Encke's  comet,  for 
example. 

*  For  when  we  know  the  length  of  the  major  axis,  we  can  find  tho  periodic  time  by 
Kepler's  law,  which  applies  as  well  to  comets  as  to  planets. 


ORBITS    AND    MOTIONS    OF    COMETS. 


243 


398.  The  elements  of  a  comet,  with  the  exception  of  its  periodic 
time,  are  calculated  in  a  manner  similar  to  those  of  the  planets. 
Three  good  observations  on  the  right  ascension  and  declination  of 
the  comet  (which  are  usually  found  by  ascertaining  its  position 
with  respect  to  certain  stars,  whose  right  ascensions  and  declina- 
tions are  accurately  known)  afford  the  means  of  calculating  these 
elements. 

The  appearance  of  the  same  comet  at  different  periods  of  its 
return  are  so  various,  (Art.  388,)  that  we  can  never  pronounce  a 
given  comet  to  be  the  same  with  one  that  has  appeared  before, 
from  any  peculiarities  in  its  physical  aspect.  The  identity  of  a 
comet  with  one  already  on  record,  is  determined  by  the  identity 
of  the  elements.  It  was  by  this  means  that  Halley  first  established 
the  identity  of  the  comet  which  bears  his  name,  with  one  that 
had  appeared  at  several  preceding  ages  of  the  world,  of  which 
so  many  particulars  were  left  on  record,  as  to  enable  him  to  cal- 
culate the  elements  at  each  period.  These  were  as  in  the  follow 
ing  table. 


Time  of  appear. 

Inclin.  of  the  orbit. 

Long,  of  the  Node. 

Long,  of  Per. 

Per.  Dist. 

Course. 

1456 
1531 
1G07 
1682 

17°  56' 
17     56 
17     02 
17    42 

48°  30' 
49     25 
50     21 

50     48 

301°  00 
301  39 
302  16 
301  36 

0.58 
0.57 
0.58 
0.58 

Retrograde. 
Retrograde. 
Retrograde. 
Retrograde. 

On  comparing  these  elements,  no  doubt  could  be  entertained 
that  they  belonged  to  one  and  the  same  body ;  and  since  the  in- 
terval between  the  successive  returns  was  seen  to  be  75  or  76 
years,  Halley  ventured  to  predict  that  it  would  again  return  in 
1758.  Accordingly,  the  astronomers  who  lived  at  that  period, 
looked  for  its  return  with  the  greatest  interest.  It  was  found 
however,  that  on  its  way  towards  the  sun  it  would  pass  very  near 
to  Jupiter  and  Saturn,  and  by  their  action  on  it,  it  would  be  re- 
tarded for  a  long  time.  Clairaut,  a  distinguished  French  mathe- 
matician, undertook  the  laborious  task  of  estimating  the  exact 
amount  of  this  retardation,  and  found  it  to  be  no  less  than  618 
days,  namely,  100  days  by  the  action  of  Jupiter,  and  518  days  by 
that  of  Saturn.  This  would  delay  its  appearance  until  early  in 
the  year  1759,  and  Clairaut  fixed  its  arrival  at  the  perihelion  within 


244  COMETS. 

a  month  of  April  13th.     It  came  to  the  perihelion  on  the  12th  of 
March. 

399.  The  return  of  Halley's  comet  in  1835,  was  looked  for  with 
no  less  interest  than  in  1759.     Several  of  the  most  accurate  math- 
ematicians of  the  age  had  calculated  its  elements  with  inconceiva- 
ble labor.     Their  zeal  was  rewarded  by  the  appearance  of  the 
expected  visitant  at  the  time  and  place  assigned ;  it  traversed  the 
northern  sky  presenting  the  very  appearances,  in  most  respects, 
that  had  been  anticipated ;  and  came  to  its  perihelion  on  the  16th 
of  November,  within  one   day    of  the  time  predicted  by  Ponte- 
coulant,  a  French  mathematician  who  had,  it  appeared,  made  the 
most   successful   calculation.*      On  its  previous   return,   it  was 
deemed  an  extraordinary  achievement  to  have  brought  the  pre- 
diction within  a  month  of  the  actual  time. 

Many  circumstances  conspired  to  render  this  return  of  Halley's 
comet  an  astronomical  event  of  transcendent  interest.  Of  all  the 
celestial  bodies,  its  history  was  the  most  remarkable ;  it  afforded 
most  triumphant  evidence  of  the  truth  of  the  doctrine  of  univer- 
sal gravitation,  and  of  course  of  the  received  laws  of  astronomy ; 
and  it  inspired  new  confidence  in  the  power  of  that  instrument, 
(the  Calculus,)  by  means  of  which  its  elements  had  been  investi- 
gated. 

400.  Encke's  comet,  by  its  frequent  returns,  affords  peculiar  fa- 
cilities for  ascertaining  the  laws  of  its  revolution ;  and  it  has  kept 
the  appointments  made  for  it,  with  great  exactness.     On  its  re- 
turn  in  1839  it  exhibited   to  the  telescope  a  globular  mass  of 
nebulous  matter,  resembling  fog,  and  moved  towards  its  perihelion 
with  great  rapidity. 

But  what  has  made  Encke's  comet  particularly  famous,  is  its 
having  first  revealed  to  us  the  existence  of  a  Resisting  Medium  in 
the  planetary  spaces.  It  has  long  been  a  question  whether  the 
earth  and  planets  revolve  in  a  perfect  void,  or  whether  a  fluid  of 
extreme  rarity  may  not  be  diffused  through  space.  A  perfect 


*  See  Professor  Loomis's  Observations  on  Halley's  Comet,  Amer.  Jour.  Science,  30 
209. 


COMETS.  245 

vacuum  was  deemed  most  probable,  because  no  such  effects  on  the 
motions  of  the  planets  could  be  detected  as  indicated  that  they  en- 
countered a  resisting  medium.  But  a  feather  or  a  lock  of  cotton 
propelled  with  great  velocity,  might  render  obvious  the  resistance 
of  a  medium  which  would  not  be  perceptible  in  the  motions  of  a 
cannon  ball.  Accordingly,  Encke's  comet  is  thought  to  have  plainly 
suffered  a  retardation  from  encountering  a  resisting  medium  in  the 
planetary  regions.  The  effect  of  this  resistance,  from  the  first  dis- 
covery of  the  comet  to  the  present  time,  has  been  to  diminish  the 
time  of  its  revolution  about  two  days.  Such  a  resistance,  by  de- 
stroying a  part  of  the  projectile  force,  would  cause  the  comet  to 
approach  nearer  to  the  sun,  and  thus  to  have  its  periodic  time 
shortened.  The  ultimate  effect  of  this  cause  will  be  to  bring  the 
comet  nearer  to  the  sun  at  every  revolution,  until  it  finally  falls 
into  that  luminary,  although  many  thousand  years  will  be  required 
to  produce  this  catastrophe.*  It  is  conceivable,  indeed,  that  the 
effects  of  such  a  resistance  may  be  counteracted  by  the  attraction 
of  one  or  more  of  the  planets  near  which  it  may  pass  in  its  succes- 
sive returns  to  the  sun. 

401.  It  is  peculiarly  interesting  to  see  a  portion  of  matter  of  a 
tenuity  exceeding  the  thinnest  fog,  pursuing  its  path  in  space,  in 
obedience  to  the  same  laws  as  those  which  regulate  such  large  and 
heavy  bodies  as  Jupiter  or  Saturn.     In  a  perfect  void,  a  speck  of 
fog  if  propelled  by  a  suitable  projectile  force  would  revolve  around 
the  sun,  and  hold  on  its  way  through  the  widest  orbit,  with  as  sure 
and  steady  a  pace  as  the  heaviest  and  largest  bodies  in  the  system. 

402.  Of  the  physical  nature  of  comets,  little  is  understood.     It  is 
usual  to  account  for  the  variations  which  their  tails  undergo  by 
referring  them  to  the  agencies  of  heat  and  cold.     The  intense  heat 
to  which  they  are  subject  in  approaching  so  near  the  sun  as  some 
of  them  do,  is  alleged  as  a  sufficient  reason  for  the  great  expansion 
of  thin  nebulous  atmospheres  forming  their  tails  ;  and  the  incon- 
ceivable cold  to  which  they  are  subject  in  receding  to  such  a  dis- 


*  Halley's  comet,  at  its  return  in  1835,  did  not  appear  to  be  affected  by  the  sup- 
posed resisting  medium,  and  its  existence  is  considered  as  still  doubtful. 


246  COMETS. 

tance  from  the  sun,  is  supposed  to  account  for  the  condensation  of 
the  same  matter  until  it  returns  to  its  original  dimensions.  Thus 
the  great  comet  of  1680  at  its  perihelion  approached  166  times 
nearer  the  sun  than  the  earth,  being  only  130,000  miles  from  the 
surface  of  the  sun.*  The  heat  which  it  must  have  received,  was 
estimated  to  be  equal  to  28,000  times  that  which  the  earth  receives 
in  the  same  time,  and  2000  times  hotter  than  red  hot  iron.  This 
temperature  would  be  sufficient  to  volatilize  the  most  obdurate 
substances,  and  to  expand  the  vapor  to  vast  dimensions ;  and  the 
opposite  effects  of  the  extreme  cold  to  which  it  would  be  subject 
in  the  regions  remote  from  the  sun,  would  be  adequate  to  condense 
it  into  its  former  volume. 

This  explanation  however,  does  not  account  for  the  direction 
of  the  tail,  extending  as  it  usually  does,  only  in  a  line  opposite  to 
the  sun.  Some  writers  therefore,  as  Delarnbre,  suppose  that  the 
nebulous  matter  of  the  comet  after  being  expanded  to  such  a  vol- 
ume, that  the  particles  are  no  longer  attracted  to  the  nucleus  un- 
less by  the  slightest  conceivable  force,  are  carried  off  in  a  direction 
from  the  sun,  by  the  impulse  of  the  solar  rays  themselves. f  But 
to  assign  such  a  power  of  communicating  motion  to  the  sun's  rays 
while  they  have  never  been  proved  to  have  any  momentum,  is 
unphilosophical ;  and  we  are  compelled  to  place  the  phenomena 
of  comets'  tails  among  the  points  of  astronomy  yet  to  be  ex- 
plained. 

403.  Since  those  comets  which  have  their  perihelion  very  near 
the  sun,  like  the  comet  of  1680,  cross  the  orbits  of  all  the  planets, 
the  possibility  that  one  of  them  may  strike  the  earth,  has  frequently 
been  suggested.  Still  it  may  quiet  our  apprehensions  on  this  sub- 
ject, to  reflect  on  the  vast  extent  of  the  planetary  spaces,  in  which 
these  bodies  are  not  crowded  together  as  we  see  them  erroneously 
represented  in  orreries  and  diagrams,  but  are  sparsely  scattered  at 
immense  distances  from  each  other.  They  are  like  insects  flying 
in  the  expanse  of  heaven.  If  a  comet's  tail  lay  with  its  axis  in  the 
plane  of  the  ecliptic  when  it  was  near  the  sun,  we  can  imagine  that 
the  tail  might  sweep  over  the  earth ;  but  the  tail  may  be  situated 

*  See  Principia,  Lib.  in,  41.  t  Delambre's  Astronomy,  t.  3,  p.  401 


COMETS.  247 

at  any  angle  with  the  ecliptic  as  well  as  in  the  same  plane  with  it, 
and  the  chances  that  it  will  not  be  in  the  same  plane,  are  almost 
infinite.  It  is  also  extremely  improbable  that  a  comet  will  cross 
the  plane  of  the  ecliptic  precisely  at  the  earth's  path  in  that  plane, 
since  it  may  as  probably  cross  it  at  any  other  point,  nearer  or 
more  remote  from  the  sun.  Still  some  comets  have  occasionally 
approached  near  to  the  earth.  Thus  Biela's  comet  in  returning  to 
the  sun  in  1832,  crossed  the  ecliptic  very  near  to  the  earth's  track, 
and  had  the  earth  been  then  at  that  point  of  its  orbit,  it  might 
have  passed  through  a  portion  of  the  nebulous  atmosphere  of  the 
comet.  The  earth  was  within  a  month  of  reaching  that  point. 
This  might  at  first  view  seem  to  involve  some  hazard  ;  yet  we  must 
consider  that  a  month  short  implied  a  distance  of  nearly  50,000,000 
miles.  La  Place  has  assigned  the  consequences  that  would  ensue 
in  case  of  a  direct  collision  between  the  earth  and  a  comet  ;*  but 
terrible  as  he  has  represented  them  on  the  supposition  that  the 
nucleus  of  the  comet  is  a  solid  body,  yet  considering  a  comet  (as 
most  of  them  doubtless  are)  as  a  mass  of  exceedingly  light  nebu- 
lous matter,  it  is  not  probable,  even  were  the  earth  to  make  its 
way  directly  through  a  comet,  that  a  particle  of  the  comet  would 
reach  the  earth.  The  portions  encountered  by  the  earth,  would 
be  arrested  by  the  atmosphere,  and  probably  inflamed  ;  and  they 
would  perhaps  exhibit  on  a  more  magnificent  scale  than  was  ever 
before  observed,  the  phenomena  of  shooting  stars,  or  meteoric 
showers. 

*  Syst.  du  Monde,  1.  iv,  c.  4. 


PART   III. — OF   THE   FIXED   STARS   AND   SYSTEM   OF   THE 
THE    WORLD. 


CHAPTER  I. 

OF   THE    FIXED    STARS CONSTELLATIONS. 

404.  THE  FIXED  STARS  are  so  called,  because,  to  common  ob- 
servation, they  always  maintain  the  same  situations  with  respect 
to  one  another. 

The  stars  are  classed,  by  their  apparent  magnitudes.  The  whole 
number  of  magnitudes  recorded  are  sixteen,  of  which  the  first  six 
only  are  visible  to  the  naked  eye  ;  the  rest  are  telescopic  stars.  As 
the  stars  which  are  now  grouped  together  under  one  of  the  first 
six  magnitudes  are  very  unequal  among  themselves,  it  has  recently 
been  proposed  to  subdivide  each  class  into  three,  making  in  all 
eighteen  instead  of  six  magnitudes  visible  to  the  naked  eye. 
These  magnitudes  are  not  determined  by  any  very  definite  scale, 
but  are  merely  ranked  according  to  their  relative  degrees  of 
brightness,  and  this  is  left  in  a  great  measure  to  the  decision  of  the 
eye  alone,  although  it  would  appear  easy  to  measure  the  compar- 
ative degree  of  light  in  a  star  by  a  photometer,  and  upon  such 
measurement  to  ground  a  more  scientific  classification  of  the  stars. 
The  brightest  stars  to  the  number  of  15  or  20  are  considered  as 
stars  of  the  first  magnitude  ;  the  50  or  60  next  brightest,  of  the 
second  magnitude  ;  the  next  200  of  the  third  magnitude  ;  and  thus 
the  number  of  each  class  increases  rapidly  as  we  descend  the  scale, 
so  that  no  less  than  fifteen  or  twenty  thousand  are  included  within 
the  first  seven  magnitudes. 

405.  The  stars  have  been  grouped  in  Constellations  from  the 
most  remote  antiquity :  a  few,  as  Orion,  Bootes,  and  Ursa  Major, 
are  mentioned  in  the  most  ancient  writings  under  the  same  names 


FIXED  STARS.  249 

as  they  bear  at  present.  The  names  of  the  constellations  are 
sometimes  founded  on  a  supposed  resemblance  to  the  objects  to 
which  the  names  belong ;  as  the  Swan  and  the  Scorpion  were  evi- 
dently so  denominated  from  their  likeness  to  those  animals  ;  but 
in  most  cases  it  is  impossible  for  us  to  find  any  reason  for  desig- 
nating a  constellation  by  the  figure  of  the  animal  or  the  hero  which 
is  employed  to  represent  it.  These  representations  were  probably 
once  blended  with  the  fables  of  pagan  mythology.  The  same  fig- 
ures, absurd  as  they  appear,  are  still  retained  for  the  convenience 
of  reference ;  since  it  is  easy  to  find  any  particular  star,  by  speci- 
fying the  part  of  the  figure  to  which  it  belongs,  as  when  we  say  a 
star  is  in  the  neck  of  Taurus,  in  the  knee  of  Hercules,  or  in  the 
tail  of  the  Great  Bear.  This  method  furnishes  a  general  clue  to 
its  position ;  but  the  stars  belonging  to  any  constellation  are  dis- 
tinguished according  to  their  apparent  magnitudes  as  follows  : — 
first,  by  the  Greek  letters,  Alpha,  Beta,  Gamma,  &c.  Thus  a 
Orionis,  denotes  the  largest  star  in  Orion,  ft  Andromeda,  the 
second  star  in  Andromeda,  and  y  Leonis,  the  third  brightest  star  in 
the  Lion.  Where  the  number  of  the  Greek  letters  is  insufficient 
to  include  all  the  stars  in  a  constellation,  recourse  is  had  to  the 
letters  of  the  Roman  alphabet,  a,  b,  c,  &c. ;  and,  in  cases  where 
these  are  exhausted,  the  final  resort  is  to  numbers.  This  is  evi- 
dently necessary,  since  the  largest  constellations  contain  many 
hundreds  or  even  thousands  of  stars.  Catalogues  of  particular 
stars  have  also  been  published  by  different  astronomers,  each 
author  numbering  the  individual  stars  embraced  in  his  list,  accord- 
ing to  the  places  they  respectively  occupy  in  the  catalogue. 
These  references  to  particular  catalogues  are  sometimes  entered 
on  large  celestial  globes.  Thus  we  meet  with  a  star  marked  84 
H.,  meaning  that  this  is  its  number  in  HerscheFs  catalogue,  or 
140  M.  denoting  the  place  the  star  occupies  in  the  catalogue  of 
Mayer. 

406.  The  earliest  catalogue  of  the  stars  was  made  by  Hippar- 
chus  of  the  Alexandrian  School,  about  140  years  before  the 
Christian  era.  A  new  star  appearing  in  the  firmament,  he  was 
induced  to  count  the  stars  and  to  record  their  positions,  in  order 
that  posterity  might  be  able  to  judge  of  the  permanency  of  the  con- 

32 


250  FIXED  STARS. 

stellations.  His  catalogue  contains  all  that  were  conspicuous  to 
the  naked  eye  in  the  latitude  of  Alexandria,  being  1022.  Most  per- 
sons unacquainted  with  the  actual  number  of  the  stars  which  com- 
pose the  visible  firmament,  would  suppose  it  to  be  much  greater  than 
this ;  but  it  is  found  that  the  catalogue  of  Hipparchus  embraces 
nearly  all  that  can  now  be  seen  in  the  same  latitude,  and  that  on 
the  equator,  when  the  spectator  has  the  northern  and  southern 
hemispheres  both  in  view,  the  number  of  stars  that  can  be  counted 
does  not  exceed  3000.  A  careless  view  of  the  firmament  in  a 
clear  night,  gives  us  the  impression  of  an  infinite  multitude  of  stars  ; 
but  when  we  begin  to  count  them,  they  appear  much  more 
sparsely  distributed  than  we  supposed,  and  large  portions  of  the 
sky  appear  almost  destitute  of  stars. 

By  the  aid  of  the  telescope,  new  fields  of  stars  present  them- 
selves of  boundless  extent ;  the  number  continually  augmenting 
as  the  powers  of  the  telescope  are  increased.  Lalande,  in  his 
Histoire  Celeste,  has  registered  the  positions  of  no  less  than 
50,000;  and  the  whole  number  visible  .in  the  largest  telescopes 
amount  to  many  millions. 

407.  It  is  strongly  recommended  to  the  learner  to  acquaint 
himself  with  the  leading  constellations  at  least,  and  with  a  few 
of  the  most  remarkable  individual  stars.  The  task  of  learning 
them  is  comparatively  easy,  and  hardly  any  kind  of  knowledge, 
attained  with  so  little  Jabor,  so  amply  rewards  the  possessor.  It 
will  generally  be  advisable,  at  the  outset,  to  get  some  one  already 
acquainted  with  the  stars,  to  point  out  a  few  of  the  most  conspicu- 
ous constellations,  those  of  the  Zodiac  for  example  :  the  learner 
may  then  resort  to  a  celestial  globe,*  and  fill  up  the  outline  by 
tracing  out  the  principal  stars  in  each  constellation  as  there  laid 
down.  By  adding  one  new  constellation  to  his  list  every  night, 
and  reviewing  those  already  acquired,  he  will  soon  become  fa- 
miliar with  the  stars,  and  will  greatly  augment  his  interest  and 
improve  his  intelligence  in  celestial  observation  and  practical  as- 
tronomy. 

*  For  the  method  of  rectifying  the  globe  so  as  to  represent  the  appearance  of  the 
heavens  on  any  particular  eve'ning,  see  page  27,  Prob.  76. 


CONSTELLATIONS.  251 


CONSTELLATIONS. 

408.  We  will  point  out  particular  marks  by  which  the  leading 
constellations  maybe  recognized,  leaving  it  to^  the  learner,  after 
he  has  found  a  constellation,  to  trace  out  additional  members  of 
it  by  the  aid  of  the  celestial  globe,  or  by  maps  of  the  stars.  Let 
us  begin  with  the  Constellations  of  the  Zodiac,  which  succeeding 
each  other  as  they  do  in  a  known  order,  are  most  easily  found. 

ARIES  (The  RAM)  is  a  small  constellation,  known  by  two  bright 
stars  which  form  his  head,  a  and  (3  Arietis.  These  two  stars  are 
four  degrees*  apart ;  and  directly  south  of  ft  at  the  distance  of  one 
degree,  is  a  smaller  star,  y  Arietis.  It  has  been  already  inti- 
mated (Art.  193,)  that  the  vernal  equinox  probably  was  near  the 
head  of  Aries,  when  the  signs  of  the  Zodiac  received  the  present 
names. 

TAURUS  (The  BULL)  will  be  readily  found  by  the  seven  stars  or 
Pleiades,  which  lie  in  his  neck.  The  largest  star  in  Taurus  is 
Aldebaran,  in  the  Bull's  eye,  a  star  of  the  first  magnitude,  of  a 
reddish  color  somewhat  resembling  the  planet  Mars.  Aldebaran 
and  four  other  stars  in  the  face  of  Taurus,  compose  the  Hyades. 

GEMINI  (The  TWINS)  is  known  by  two  very  bright  stars,  Castor 
and  Pollux,  five  degrees  asunder.  Castor  (the  northern)  is  of  the 
first,  and  Pollux  of  the  second  magnitude. 

CANCER  (The  CRAB).  There  are  no  large  stars  in  this  constel- 
lation, and  it  is  regarded  as  less  remarkable  than  any  other  in  the 
Zodiac.  It  contains  however  an  interesting  group  of  small  stars, 
called  Prcesepe  or  the  Nebula  of  Cancer,  which  resembles  a  comet, 
and  is  often  mistaken  for  one,  by  persons  unacquainted  with 
the  stars.  With  a  telescope  of  very  moderate  powers  this  nebula 
is  converted  into  a  beautiful  assemblage  of  exceedingly  bright  stars. 

LEO  (The  LION)  is  a  very  large  constellation,  and  has  many 
interesting  members.  Regulus  (a  Leonis)  is  a  star  of  the  first 
magnitude,  which  lies  directly  in  the  ecliptic,  and  is  much  used  in 
astronomical  observations.  North  of  Regulus  lies  a  semi-circle  of 
bright  stars,  forming  a  sickle  of  which  Regulus  is  the  handle. 

*  These  measures  are  not  intended  to  be  stated  with  exactness,  but  only  with  such 
a  degree  of  accuracy  as  may  serve  for  a  general  guide. 


252  FIXED   STARS. 

Denebola,  a  star  of  the  second  magnitude,  is  in  the  Lion's  tail,  25° 
northeast  of  Regulus. 

VIRGO  (The  VIRGIN)  extends  a  considerable  way  from  west 
to  east,..but  contains  only  a  few  bright  stars.  Spica,  however,  is 
a  star  of  the  firsts magnitude,  and  lies  a  little  east  of  the  place  of 
the  autumnal  equinox.  Eighteen  degrees  eastward  of  Denebola, 
and  twenty  degrees  north  of  Spica,  is  Vindemiatrix,  in  the  arm  of 
Virgo,  a  star  of  the  third  magnitude. 

LIBRA  (The  BALANCE)  is  distinguished  by  three  large  stars,  of 
which  the  two  brightest  constitute  the  beam  of  the  balance,  and 
the  smallest  forms  the  top  or  handle. 

SCORPIO  (The  SCORPION)  is  one  of  the  finest  of  the  constella- 
tions. His  head  is  formed  of  five  bright  stars  arranged  in  the 
arc  of  a  circle,  which  is  crossed  in  the  center  by  the  ecliptic  nearly 
at  right  angles,  near  the  brightest  of  the  five,  fS  Scorpionis.  Nine 
degrees  southeast  of  this,  is  a  remarkable  star  of  the  first  magni- 
tude, of  a  reddish  color,  called  Cor  Scorpionis.  or  Antares.  South 
of  this  a  succession  of  bright  stars  sweep  round  towards  the  east, 
terminating  in  several  small  stars,  forming  the  tail  of  the  Scorpion. 

SAGITTARIUS  (The  ARCHER).  Northeast  of  the  tail  of  the  Scor- 
pion, are  three  stars  in  the  arc  of  a  circle  which  constitute  the  bow 
of  the  Archer,  the  central  star  being  the  brightest,  directly  west 
of  which  is  a  bright  star  which  forms  the  arrow. 

CAPRICORNUS  (The  GOAT)  lies  northeast  of  Sagittarius,  and  is 
known  by  two  bright  stars,  three  degrees  apart,  which  form  the 
head. 

AQUARIUS  (The  WATER  BEARER)  is  recognized  by  two  stars 
in  a  line  with  a  Capricorn^  forming  the  shoulders  of  the  figure. 
These  two  stars  are  10°  apart,  and  3°  southeast  is  a  third  star, 
which  together  with  the  other  two,  makes  an  acute  triangle,  of 
which  the  westernmost  is  the  vertex. 

PISCES  (The  FISHES)  lie  between  Aquarius  and  Aries.  They 
are  not  distinguished  by  any  large  stars,  but  are  connected  by  a 
series  of  small  stars,  that  form  a  crooked  line  between  them. 
Piscis  Australis,  the  Southern  Fish,  lies  directly  below  Aquarius, 
and  is  known  by  a  singfe  bright  star  far  in  the  south,  having  a 
declination  of  30°.  The  name  of  this  star  is  Fomalhaut,  and  is 
much  used  in  astronomical  measurements. 


CONSTELLATIONS.  253 

409.  The  Constellations  of  the  Zodiac,  being  first  well  learned, 
so  as  to  be  readily  recognized,  will  facilitate  the  learning  of  others 
that  lie  north  and  south  of  them.  Let  us  therefore  next  review 
the  principal  Northern  Constellations,  beginning  north  of  Aries  and 
proceeding  from  west  to  east. 

ANDROMEDA,  is  characterized  by  three  stars  of  the  second  mag- 
nitude, situated  in  a  straight  line,  extending  from  west  to  east. 
The  middle  star  is  about  17°  north  of  /3  Arietis.  It  is  in  the  girdle 
of  Andromeda,  and  is  named  Mirach.  The  other  two  lie  at  about 
equal  distances,  14C  west  and  east  of  Mirach.  The  western  star, 
in  the  head  of  Andromeda,  lies  in  the  Equinoctial  Colure.  The 
eastern  star,  Almaak,  is  situated  in  the  foot. 

PERSEUS  lies  directly  north  of  the  Pleiades,  and  contains  sev- 
eral bright  stars.  About  18°  from  the  Pleiades  is  Algol,  a  star 
of  the  second  magnitude,  in  the  Head  of  Medusa,  which  forms  a 
part  of  the  figure ;  and  9°  northeast  of  Algol  is  Algenib,  of  the 
same  magnitude  in  the  back  of  Perseus.  Between  Algenib  and 
the  Pleiades  are  three  bright  stars,  at  nearly  equal  intervals,  which 
compose  the  right  leg  of  Perseus. 

AURIGA  (the  WAGONER)  lies  directly  east  of  Perseus,  and  extends 
nearly  parallel  to  that  constellation  from  north  to  south.  Capella, 
a  very  white  and  beautiful  star  of  the  first  magnitude,  distinguishes 
this  constellation.  The  feet  of  Auriga  are  near  the  Bull's  Horns. 

The  LYNX  comes  next,  but  presents  nothing  particularly  inter- 
esting, containing  no  stars  above  the  fourth  magnitude. 

LEO  MINOR  consists  of  a  collection  of  small  stars  north  of  the 
sickle  in  Leo,  and  south  of  the  Great  Bear.  Its  largest  star  is 
only  of  the  third  magnitude. 

COMA  BERENICES  is  a  cluster  of  small  stars,  north  of  Denebola, 
in  the  tail  of  the  lion,  and  of  the  head  of  Virgo.     About   12 
directly  north  of  Berenice's  Hair,  is  a  single  bright  star  called  Cor 
Caroli,  or  Charles's  Heart. 

BOOTES,  which  comes  next,  is  easily  found  by  means  of  Arc- 
turus,  a  star  of  the  first  magnitude,  of  a  reddish  color,  which  is 
situated  near  the  knee  of  the  figure.  Arcturus  is  accompanied 
by  three  small  stars  forming  a  triangle  a  little  to  the  southwest. 
Two  bright  stars  y  and  5  Bootis,  form  the  shoulders,  and  /3  of  the 
third  magnitude  is  in  the  head  of  the  figure. 


254  FIXED    STARS. 

CORONA  BOREALIS  (The  CROWN)  which  is  situated  E.  of  Bootes, 
is  very  easily  recognized,  composed  as  it  is  of  a  semi-circle  of 
bright  stars.  In  the  center  of  the  bright  crown,  is  a  star  of  the 
second  magnitude,  called  gemma ;  the  remaining  stars  are  all  much 
smaller. 

HERCULES,  lying  between  the  Crown  on  the  west  and  the  Lyre 
on  the  east,  is  very  thickly  set  with  stars,  most  of  which  are  quite 
small.  This  Constellation  covers  a  great  extent  of  the  sky,  es- 
pecially from  N.  to  S.,  the  head  terminating  within  15°  of  the 
equator,  and  marked  by  a  star  of  the  third  magnitude,  called  Ras- 
algethi,  which  is  the  largest  in  the  Constellation. 

OPHIUCHUS  is  situated  directly  south  of  Hercules,  extending  some 
distance  on  both  sides  of  the  equator,  the  feet  resting  on  the  Scor- 
pion. The  head  terminates  near  the  head  of  Hercules,  and  like 
that,  is  marked  by  a  bright  star  within  5°  of  a  Herculis.  Ophiu- 
chus  is  represented  as  holding  in  his  hands  the  SERPENT,  the  head 
of  which,  consisting  of  .three  bright  stars,  is  situated  a  little  south 
of  the  Crown.  The  folds  of  the  serpent  will  be  easily  followed 
by  a  succession  of  bright  stars  which  extend  a  great  way  to  the 
east. 

AQUILA  (The  EAGLE)  is  conspicuous  for  three  bright  stars  in  its 
neck,  of  which  the  central  one,  Altair,  is  a  very  brilliant  white 
star  of  the  first  magnitude.  Antinous  lies  directly  south  of  the 
Eagle,  and  north  of  the  head  of  Capricornus. 

DELPHINUS  (The  DOLPHIN)  is  a  small  but  beautiful  Constellation, 
a  few  degrees  east  of  the  Eagle,  and  is  characterized  by  four  bright 
stars  near  to  one,  another,  forming  a  small  rhombic  square.  An- 
other star  of  the  same  magnitude  5°  south,  makes  the  tail. 

PEGASUS  lies  between  Aquarius  on  the  southwest  and  Andromeda 
on  the  northeast.  It  contains  but  few  large  stars.  A  very  regu- 
lar square  of  bright  stars  is  composed  of  a  Andromedce,  and  the 
three  largest  stars  in  Pegasus,  namely,  Scheat,  Markab,  and  Alge- 
nib.  The  sides  composing  this  square  are  each  about  15°.  Alge- 
nib  is  situated  in  the  equinoctial  colure. 

410.  We  may  now  review  the  Constellations  which  surround 

the  North  Pole,  within  the  circle  of  perpetual  apparition.     (Art.  54.) 

URSA  MINOR  (The  LITTLE  BEAR)  lies  nearest  the  pole.     The 


CONSTELLATIONS.  255 

Pole-star,  Polaris,  is  in  the  extremity  of  the  tail,  and  is  of  the  third 
magnitude.  Three  stars  in  a  straight  line  4°  or  5°  apart,  com- 
mencing with  the  Pole-star,  lead  to  a  trapezium  of  four  stars,  and 
the  whole  seven  form  together  a  dipper,  the  trapezium  being  the 
body,  and  the  three  stars  the  handle. 

URSA  MAJOR  (The  GREAT  BEAR)  is  situated  between  the  pole 
and  the  Lesser  Lion,  and  is  usually  recognized  by  the  figure  of  a 
larger  and  more  perfect  dipper,  which  constitutes  the  hinder  part 
of  the  animal.  This  has  also  seven  stars,  four  in  the  body  of  the 
dipper,  and  three  in  the  handle.  All  these  are  stars  of  much  ce- 
lebrity. The  two  in  the  western  side  of  the  dipper,  a  and  j3,  are 
called  Pointers,  on  account  of  their  always  being  in  a  right  line 
with  the  Pole-star,  and  therefore  affording  an  easy  mode  of  finding 
that.  The  first  star  in  the  tail,  next  the  body,  is  named  Alioth,  and 
the  second  Mizar.  The  head  of  the  Great  Bear  lies  far  to  the 
westward  of  the  Pointers,  and  is  composed  of  numerous  small 
stars  ;  and  the  feet  are  severally  composed  of  two  small  stars  very 
near  to  each  other. 

DRACO  (The  DRAGON)  winds  round  between  the  Great  and  Lit- 
tle Bear ;  and  commencing  with  the  tail,  between  the  Pointers  and 
the  Pole-star,  it  is  easily  traced  by  a  succession  of  bright  stars  ex- 
tending from  west  to  east ;  passing  under  Ursa  Minor,  it  returns 
westward,  and  terminates  in  a  triangle  which  forms  the  head  of 
Draco,  near  the  feet  of  Hercules,  northwest  of  Lyra. 

CEPHEUS  lies  eastward  of  the  breast  of  the  Dragon,  but  has  no 
stars  above  the  third  magnitude. 

^CASSIOPEIA  is  known  by  the  figure  of  a  chair,  composed  of  four 
stars  which  form  the  legs,  and  two  which  form  the  back.  This 
Constellation  lies  between  Perseus  and  Cepheus,  in  the  Milky. 
Way. 

CYGNUS  (The  SWAN)  is  situated  also  in  the  Milky  Way,  some 
distance  southwest  of  Cassiopeia,  towards  the  Eagle.  Three 
bright  stars,  which  lie  along  the  Milky  Way,  form  the  body  and 
neck  of  the  Swan,  and  two  others  in  a  line  with  the  middle  one  of 
the  three,  one  above  and  one  below,  constitute  the  wings.  This 
Constellation  is  among  the  few  that  exhibit  some  resemblance  to 
the  animals  whose  names  they  bear. 

LYRA  (The  LYRE)  is  directly  west  of  the  Swan,  and  is  easily 


256  FIXED  STARS. 

distinguished  by  a  beautiful  white  star  of  the  first  magnitude,  a 
Lyrce. 

411.  The  Southern    Constellations  are  comparatively   few   in 
number.     We  shall  notice  only  the  Whale,  Orion,  the  Greater  and 
Lesser  Dog,  Hydra,  and  the  Crow. 

CETUS  (The  WHALE)  is  distinguished  rather  for  its  extent  than  its 
brilliancy,  reaching  as  it  does  through  40°  of  longitude,  while  none 
of  its  stars  except  one,  are  above  the  third  magnitude.  Menkar  (a 
Ceti)  in  the  mouth,  is  a  star  of  the  second  magnitude,  and  several 
other  bright  stars  directly  south  of  Aries,  make  the  head  and  neck 
of  the  Whale.  Mira  (o  Ceti)  in  the  neck  of  the  Whale,  is  a  varia- 
ble star. 

ORION  is  one  of  the  largest  and  most  beautiful  of  the  constella- 
tions, lying  southeast  of  Taurus.  A  cluster  of  small  stars  form  the 
head  ;  two  large  stars,  Betalgeus  of  the  first  and  Bellatrix  of  the 
second  magnitude,  make  the  shoulders ;  three  more  bright  stars 
compose  the  buckler,  and  three  the  sword ;  and  Rigel,  another 
star  of  the  first  magnitude,  makes  one  of  the  feet.  In  this  Con- 
stellation there  are  70  stars  plainly  visible  to  the  naked  eye,  inclu- 
ding two  of  the  first  magnitude,  four  of  the  second,  and  three  of 
the  third. 

CANIS  MAJOR  lies  S.  E.  of  Orion,  and  is  distinguished  chiefly  by 
its  containing  the  largest  of  the  fixed  stars,  Sirius. 

CANIS  MINOR,  a  little  north  of  the  equator,  between  Canis  Major 
and  Gemini,  is  a  small  Constellation,  consisting  chiefly  of  two 
stars,  of  which  Procyon  is  of  the  first  magnitude. 

HYDRA  has  its  head  near  Procyon,  consisting  of  a  number  of 
stars  of  ordinary  brightness.  About  15°  S.  E.  of  the  head,  is  a 
star  of  the  second  magnitude,  forming  the  heart,  (Cor  Hydras) ; 
and  eastward  of  this,  is  a  long  succession  of  stars  of  the  fourth  and 
fifth  magnitudes  composing  the  body  and  the  tail,  and  reaching  a 
few  degrees  south  of  Spica  Virginis. 

CORVUS  (The  CROW)  is  represented  as  standing  on  the  tail  of 
Hydra.  It  consists  of  small  stars,  only  three  of  which  are  as 
large  as  the  third  magnitude. 

412.  The  foregoing  brief  sketch  is  designed  merely  to  aid  the 


CLUSTERS  OF  STARS.  257 

student  in  finding  the  principal  constellations  and  the  largest  fixed 
stars.  When  we  have  once  learned  to  recognize  a  constellation 
by  some  characteristic  marks,  we  may  afterwards  fill  up  the  out- 
line by  the  aid  of  a  celestial  globe  or  a  map  of  the  stars.  It  will  be 
of  little  avail,  however,  merely  to  commit  this  sketch  to  memory  ; 
but  it  will  be  very  useful  for  the  student  at  once  to  render  himself 
familiar  with  it,  from  the  actual  specimens  which  every  clear 
evening  presents  to  his  view. 


CHAPTER    II. 

• 

CLUSTERS   OF    STARS NEBULAE VARIABLE    STARS TEMPORARY 

STARS DOUBLE  STARS. 

413.  IN  various  parts  of  the  firmament  are  seen  large  groups  or 
clusters,  which,  either  by  the  naked  eye,  or  by  the  aid  of  the  small- 
est telescope,  are  perceived  to  consist  of  a  great  number  of  small 
stars.  Such  are  the  Pleiades,  Coma  Berenices,  and  Prsesepe  or 
the  Bee-hive,  in  Cancer.  The  Pleiades,  or  Seven  Stars,  as  they 
are  called,  in  the  neck  of  Taurus,  is  the  most  conspicuous  cluster. 
When  we  look  directly  at  this  group,  we  cannot  distinguish 
more  than  six  stars,  but  by  turning  the  eye  sideways*  upon  it,  we 
discover  that  there  are  many  more.  Telescopes  show  50  or  60 
stars  crowded  together  and  apparently  insulated  from  the  other 
parts  of  the  heavens.f  Coma  Berenices  has  fewer  stars,  but  they 
are  of  a  larger  class  than  those  which  compose  the  Pleiades.  The 
Bee-hive  or  Nebula  of  Cancer  as  it  is  called,  is  one  of  the  finest 
objects  of  this  kind  for  a  small  telescope,  being  by  its  aid  con- 
verted into  a  rich  congeries  of  shining  points.  The  head  of  Orion 
affords  an  example  of  another  cluster,  though  less  remarkable  than 
the  others. 

*  Indirect  vision  is  far  more  delicate  than  direct.  Thus  we  can  see  the  Zodiacal 
Light  or  a  Comet's  Tail,  much  more  distinctly  and  better  defined,  if  we  fix  one  eye  on 
a  part  of  the  heavens  at  some  distance,  and  turn  the  other  eye  obliquely  upon  the  ob- 
ject. tSirJ.Herschel. 

33 


258  FIXED  STARS. 

414.  Nebula  are  those  faint  misty  appearances  which  resemble 
comets,  or  a  small  speck  of  fog.     The  Galaxy  or  Milky  Way  pre- 
sents a  continued  succession  of  large  nebulae.     A  very  remarkable 
Nebula,  visible  to  the  naked  eye,  is  seen  in  the  girdle  of  Androme- 
da.    No  powers  of  the  telescope  have  been  able  to  resolve  this 
into  separate  stars.     Its  dimensions  are  astonishingly  great.     In 
diameter  it  is  about  15'.     The  telescope  reveals  to  us  innumerable 
objects  of  this  kind.     Sir  William  Herschel  has  given  catalogues 
of  2000  Nebulae,  and  has  shown  that  the  nebulous  matter  is  dis- 
tributed through  the  immensity  of  space  in  quantities  inconceiva- 
bly great,  and  in  separate  parcels  of  all  shapes  and  sizes,  and  of 
all  degrees  of  brightness  between  a  mere  milky  appearance  and 
the  condensed  light  of  a  fixed  star.     Finding  that  the  gradations 
between  the  two  extremes  were  tolerably  regular,  he  thought  it 
probable  that  the  nebulae  form  the  materials  out  of  which  nature 
elaborates  suns  and  systems ;  and  he  conceived  that,  in  virtue  of 
a  central  gravitation,  each  parcel  of  nebulous  matter  becomes 
more  and  more  condensed,  and  assumes  a  rounder  form.     He  in- 
fers from  the  eccentricity  of  its  shape,  and  the  effects  of  the  mutual 
gravitation*  of  its  particles,  that  it  acquires  gradually  a  rotary  mo- 
tion ;  that  the  condensation   goes  on  increasing  until  the  mass- 
acquires  consistency  and  solidity,  and  all  the  other  characters  of  a 
comet  or  a  planet ;  that  by  a  still  further  process  of  condensation, 
the  body  becomes  a  real  star,  self-shining  ;  and  that  thus  the  waste 
of  the  celestial  bodies,  by  the  perpetual  diffusion  of  their  light,  is 
continually  compensated  and  restored  by  new  formations  of  such 
bodies,  to  replenish  forever  the  universe  with  planets  and  stars.* 

415.  These  opinions  are  recited  here  rather  out  of  respect  to 
their  notoriety  and  celebrity,  than  because  we  suppose  them  to  be 
founded  on  any  better  evidence  than  conjecture.     The  Philosophi- 
cal Transactions  for  many  years,  both  before  and  after  the  com- 
mencement of  the  present  century,  abound   with  both  the  ob- 
servations and  speculations  of  Sir  William  Herschel.     The  for- 
mer are   deserving  of  all  praise ;  the  latter   of  less  confidence. 
Changes,  however,  are  going  on  in  some  of  the  nebulae,  which 

*  Phil,  Trans.  1811. 


NEBULAE.  259 

plainly  show  that  they  are  not,  like  planets  and  stars,  fixed  and 
permanent  creations.  Thus  the  great  nebula  in  the  girdle  of  An- 
dromeda, has  very  much  altered  its  structure  since  it  first  became 
an  object  of  telescopic  observation.*  Many  of  the  nebulae  are  of 
a  globular  form,  (Fig.  72,  a)  but  frequently  they  present  the  ap- 

(Fig.  72,  a.)  (Fig.72,  6.) 


pearance  of  a  rapid  increase  of  numbers  towards  the  center,  (Fig. 
72,  b)  the  exterior  boundary  being  irregular,  and  the  central  parts 
more  nearly  spherical. 

416.  The  Nebula  in  the  sword  of  Orion  is  particularly  celebrated, 
being  very  large  and  of  a  peculiarly  interesting  appearance.f  Ac- 
cording to  Sir  John  Herschel,  its  nebulous  character  is  very  dif- 
ferent from  what  might  be  supposed  to  arise  from  the  assemblage 
of  an  immense  collection  of  small  stars.  It  is  formed  of  little 
flocculent  masses  like  wisps  of  clouds  ;  and  such  wisps  seem  to 
adhere  to  many  small  stars  at  its  outskirts,  and  especially  to  one 
considerable  star  which  it  envelops  with  a  nebulous  atmosphere  of 
considerable  extent  and  singular  figure. 

Descriptions,  however,  can  convey  but  a  very  imperfect  idea 
of  this  wonderful  class  of  astronomical  objects,  and  we  would 
therefore  urge  the  learner  studiously  to  avail  himself  of  the  first 
opportunity  he  may  have  to  view  them  through  a  large  telescope, 
especially  the  Nebula  of  Andromeda  and  of  Orion. 


Nebulous  Stars  are  such  as  exhibit  a  sharp  and  brilliant 
star  surrounded  by  a  disk  or  atmosphere  of  nebulous  matter. 
These  atmospheres  in  some  cases  present  a  circular,  in  others  an 
oval  figure  ;  and  in  some  instances,  the  nebula  consists  of  a  long, 
narrow  spindle-shaped  ray,  tapering  away  at  both  ends  to  points^ 

«  Astron.  Trans.  II,  495.  t  See  Article  V.  of  the  Addenda. 


260  FIXED  STARS. 

Annular  Nebulce  also  exist,  but  are  among  the  rarest  objects  in 
the  heavens.  The  most  conspicuous  of  this  class,  is  to  be  found 
exactly  half  way  between  the  stars  (3  and  7  Lyrae,  and  may  be  seen 
with  a  telescope  of  moderate  power.* 

Planetary  Nebula  constitute  another  variety,  and  are  very  re- 
markable objects.  They  have,  as  their  name  imports,  exactly  the 
appearance  of  planets.  Whatever  may  be  their  nature,  they  must 
be  of  enormous  magnitude.  One  of  them  is  to  be  found  in  the 
parallel  of  v  Aquarii,  and  about  5m.  preceding  that  star.  Its  appa- 
rent diameter  is  about  20".  Another  in  the  Constellation  An- 
dromeda, presents  a  visible  disk  of  12",  perfectly  defined  and 
round.  Granting  these  objects  to  be  equally  distant  from  us  with 
the  stars,  their  real  dimensions  must  be  such  as,  on  the  lowest 
computation,  would  fill  the  orbit  of  Uranus.  It  is  no  less  evident 
that,  if  they  be  solid  bodies,  of  a  solar  nature,  the  intrinsic  splendor 
of  their  surfaces  must  be  almost  infinitely  inferior  to  that  of  the 
sun.  A  circular  portion  of  the  sun's  disk,  subtending  an  angle  of 
20",  would  give  a  light  equal  to  100  full  moons;  while  the  objects 
in  question  are  hardly,  if  at  all,  discernible  with  the  naked  eye.f 

418.  The  Galaxy  or  Milky  Way  is  itself  supposed  by  some  to 
be  a  nebula  of  which  the  sun  forms  a  component  part ;  and  hence 
it  appears  so  much  greater  than  other  nebulae  only  in  consequence 
of  our  situation  with  respect  to  it,  and  its  greater  proximity  to  our 
system.     So  crowded  are  the  stars  in  some  parts  of  this  zone,  that 
Sir  William  Herschel,  by  counting  the  stars  in  a  single  field  of  his 
telescope,  estimated  that  50,000  had  passed  under  his  review  in  a 
zone  two  degrees  in  breadth  during  a  single  hour's  observation. 
Notwithstanding  the  apparent  contiguity  of  the  stars  which  crowd 
the  galaxy,  it  is  certain  that  their  mutual  distances  must  be  incon- 
ceivably great. 

419.  VARIABLE  STARS  are  those  which  undergo  a  periodical 


*  A  list  of  288  bright  nebulas,  with  references  to  well  known  stars,  near  which  they 
are  situated,  is  given  in  the  Edinburg  Enyclopaedia,  Art.  Astronomy,  p.  781.  It  is  con- 
venient  for  finding  any  required  nebula, 

t  Sir  J.  Herschel. 


TEMPORARY  STARS.  261 

change  of  brightness.  One  of  the  most  remarkable  is  the  star 
Mira  in  the  Whale,  (o  Ceti.)  It  appears  once  in  1 1  months,  re- 
mains at  its  greatest  brightness  about  a  fortnight,  being  then,  on 
some  occasions,  equal  to  a  star  of  the  second  magnitude.  It  then 
decreases  about  three  months,  until  it  becomes  completely  invisible, 
and  remains  so  about  five  months,  when  it  again  becomes  visible, 
and  continues  increasing  during  the  remaining  three  months  of  its 
period. 

Another  very  remarkable  variable  star  is  Algol  (/3  Persei.)  It 
is  usually  visible  as  a  star  of  the  second  magnitude,  and  continues 
such  for  2d.  14h.  when  it  suddenly  begins  to  diminish  in  splendor, 
and  in  about  3i  hours  is  reduced  to  the  fourth  magnitude.  It  then 
begins  again  to  increase,  and  in  3?  hours  more,  is  restored  to  its 
usual  brightness,  going  through  all  its  changes  in  less  than  three 
days.  This  remarkable  law  of  variation  appears  strongly  to  sug- 
gest the  revolution  round  it  of  some  opake  body,  which,  when  in- 
terposed between  us  and  Algol,  cuts  off  a  large  portion  of  its  light. 
It  is  (says  Sir  J.  Herschel)  an  indication  of  a  high  degree  of  ac- 
tivity in  regions  where,  but  for  such  evidences,  we  might  conclude 
all  lifeless.  Our  sun  requires  almost  nine  times  this  period  to  per- 
form a  revolution  on  its  axis.  On  the  other  hand,  the  periodic 
time  of  an  opake  revolving  body,  sufficiently  large,  which  would 
produce  a  similar  temporary  obscuration  of  the  sun,  seen  from  a 
fixed  star,  would  be  less  than  fourteen  hours. 

The  duration  of  these  periods  is  extremely  various.  While  that 
of  ]8  Persei  above  mentioned,  is  less  than  three  days,  others  are 
more  than  a  year,  and  others  many  years. 

420.  TEMPORARY  STARS  are  new  stars  which  have  appeared 
suddenly  in  the  firmament,  and  after  a  certain  interval,  as  suddenly 
disappeared  and  returned  no  more. 

It  was  the  appearance  of  a  new  star  of  this  kind  125  years  be- 
fore the  Christian  era,  that  prompted  Hipparchus  to  draw  up  a 
catalogue  of  the  stars,  the  first  on  record.  Such  also  was  the  star 
which  suddenly  shone  out  A.  D.  389,  in  the  Eagle,  as  bright  as 
Venus,  and  after  remaining  three  weeks,  disappeared  entirely. 
At  other  periods,  at  distant  intervals,  similar  phenomena  have  pre- 
sented themselves.  Thus  the  appearance  of  a  star  in  1572,  was 


262  FIXED    STARS. 

so  sudden,  that  Tycho  Brahe  returning  home  one  day  was  sur- 
prized to  find  a  collection  of  country  people  gazing  at  a  star,  which 
he  was  sure  did  not  exist  half  an  hour  before.  It  was  then  as 
bright  as  Sirius,  and  continued  to  increase  until  it  surpassed  Jupi- 
ter when  brightest,  and  was  visible  at  mid-day.  In  a  month  it 
began  to  diminish,  and  in  three  months  afterwards  it  had  entirely 
disappeared. 

It  has  been  supposed  by  some  that  in  a  few  instances,  the  same 
star  has  returned,  constituting  one  of  the  periodical  or  variable 
stars  of  a  long  period. 

Moreover,  on  a  careful  re-examination  of  the  heavens,  and  a 
comparison  of  catalogues,  many  stars  are  now  found  to  be  miss- 
ing.* 

421.  DOUBLE  STARS  are  those  which  appear  single  to  the  naked 
eye,  but  are  resolved  into  two  by  the  telescope ;  or  if  not  visible 
to  the  naked  eye,  are  seen  in  the  telescope  very  close   together. 
Sometimes  three  or  more  stars  are  found  in  this  near  connexion, 
constituting  triple  or  multiple  stars.     Castor,  for  example,  when 
seen  by  the  naked  eye,  appears  as  a  single  star,  but  in  a  telescope 
even  of  moderate  powers,  it  is  resolved  into  two  stars  of  between 
the  third  and  fourth  magnitudes,  within  5"  of  each  other.     These 
two  stars  are  nearly  of  equal  size,  but  frequently  one  is  exceedingly 
small  in  comparison  with  the  other,  resembling  a  satellite  near  its 
primary,  although  in  distance,  in  light,  and  in  other  characteristics, 
each  has  all  the  attributes  of  a  star,  and  the  combination  therefore 
cannot  be  that  of  a  planet  with  a  satellite.     In  some  instances,  also, 
the  distance  between  these  objects  is  much  less  than  5",  and  in 
many  cases  it  is  less  than  1".     The  extreme   closeness,  together 
with  the  exceeding  minuteness  of  most  of  the  double  stars,  requires 
the  best  telescope  united  with  the  most  acute  powers  of  observa- 
tion.    Indeed,  certain  of  these  objects  are  regarded  as  the  severest 
tests  both  of  the  excellence  of  the  instrument,  and  of  the  skill  of  the 
observer. 

422.  Many  of  the  double  stars  exhibit  the  curious  and  beautiful 

*SirJ.  Herschel. 


DOUBLE    STARS.  263 

phenomena  of  contrasted  or  complementary  colors.*  In  such  in- 
stances, the  larger  star  is  usually  of  a  ruddy  or  orange  hue,  while 
the  smaller  one  appears  blue  or  green,  probably  in  virtue  of  that 
general  law  of  optics,  which  provides  that  when  the  retina  is  ex- 
cited by  any  bright  colored  light,  feebler  lights  which  seen  alone 
would  produce  no  sensation  but  of  whiteness,  appear  colored,  with 
the  tint  complementary  to  that  of  the  brighter.  Thus  a  yellow 
color  predominating  in  the  light  of  the  brighter  star,  that  of  the 
less  bright  one  in  the  same  field  of  view  will  appear  blue  ;  while, 
if  the  tint  of  the  brighter  star  verges  to  crimson,  that  of  the  other 
will  exhibit  a  tendency  to  green,  or  even  under  favorable  circum- 
stances, will  appear  as  a  vivid  green.  The  former  contrast  is 
beautifully  exhibited  by  »  Cancri,  the  latter  by  y  Andromedae,  both 
fine  double  stars.  If,  however,  the  colored  star  is  much  the 
less  bright  of  the  two,  it  will  not  materially  affect  the  other.  Thus 
for  instance,  t\  Cassiopeiae  exhibits  the  beautiful  combination  of  a 
large  white  star,  and  a  small  one  of  a  rich  ruddy  purple.  It  is  by 
no  means,  however,  intended  to  say,  that  in  all  such  cases,  one  of 
the  colors  is  the  mere  effect  of  contrast,  and  it  may  be  easier  sug- 
gested in  words,  than  conceived  in  imagination,  what  variety  of 
illumination  two  suns,  a  red  and  green,  or  a  yellow  and  a  blue  sun, 
must  afford  a  planet  circulating  about  either ;  and  what  charming 
contrasts  and  "  grateful  vicissitudes,"  a  red  and  green  day  for  in- 
stance, alternating  with  a  white  one  and  with  darkness,  might  arise 
from  the  presence  or  absence  of  one  or  other,  or  both  above  the 
horizon.  Insulated  stars  of  a  red  color,  almost  as  deep  as  that  of 
blood,  occur  in  many  parts  of  the  heavens,  but  no  green  or  blue  star, 
of  any  decided  hue,  has,  we  believe,  ever  been  noticed,  unassociated 
with  a  companion  brighter  than  itself.t 

423.  Our  knowledge  of  the  double  stars  almost  commenced  with 
Sir  William  Herschel,  about  the  year  1780.  At  the  time  he 

*  Complementary  colors  are  such  as  together  make  white  light.  If  all  the  colors  of 
the  spectrum  be  laid  down  on  a  circular  ring,  each  occupying  its  proportionate  space,  any 
two  colors  on  the  opposite  sides  of  the  zone,  are  complementary  to  each  other,  and 
when  of  the  same  degree  of  intensity,  they  compose  white  light,  firewater's  Optics, 
Part  in,  c.  26. 

t  Sir  J.  HerscheL 


264  FIXED    STARS. 

began  his  search  for  them,  he  was  acquainted  with  only  four. 
Within  five  years,  he  discovered  nearly  700  double  stars.*  In  his 
memoirs  published  in  the  Philosophical  Transactions^  he  gave 
most  accurate  measurements  of  the  distances  between  the  two 
stars,  and  of  the  angle  which  a  line  joining  the  two,  formed  with 
the  parallel  of  declination.J  These  data  would  enable  him,  or  at 
least  posterity,  to  judge  whether  these  minute  bodies  ever  change 
their  position  with  respect  to  each  other. 

Since  1821,  these  researches  have  been  prosecuted  with  great 
zeal  and  industry  by  Sir  James  South  and  Sir  John  Herschel  in 
England,  and  by  Professor  Struve  at  Dorpat  in  Russia,  and  the 
whole  number  of  double  stars  now  known,  amounts  to  several 
thousands.§  Two  circumstances  add  a  high  degree  of  interest 
to  the  phenomena  of  the  double  stars, — the  first  is,  that  a  few  of 
them  at  least  are  found  to  have  a  revolution  around  each  other,  and 
the  second,  that  they  are  supposed  to  afford  the  means  of  obtain- 
ing the  parallax  of  the  fixed  stars.  Of  these  topics  we  shall  treat 
in  the  next  chapter. 


CHAPTER    III. 

MOTIONS    OF    THE    FIXED    STARS DISTANCES NATURE. 

424.  IN  1803,  Sir  William  Herschel  first  determined  and  an- 
nounced to  the  world,  that  there  exist  among  the  stars,  separate 
systems,  composed  of  two  stars  revolving  about  each  other  in  regu- 
lar orbits.  These  he  denominated  Binary  Stars,  to  distinguish 
them  from  other  double  stars  where  no  such  motion  is  detected, 
and  whose  proximity  to  each  other  may  possibly  arise  from  casual 
'uxta-position,  or  from  one  being  in  the  range  of  the  other.  Be- 

*  During  his  life  he  observed  in  all,  2400  double  stars. 

t  Phil.  Trans.  1782—1785.  t  Baily,  Astron.  Trans.  11.  542. 

$  The  Catologue  of  Struve,  contains  3063. 


MOTIONS    OF    THE    FIXED    STARS. 


265 


tween  fifty  and  sixty  instances  of  changes  to  a  greater  or  less 
amount  of  the  relative  position  of  double  stars,  are  mentioned  by 
Sir  William  Herschel ;  and  a  few  of  them  had  changed  their  places 
so  much  within  25  years,  and  in  such  order,  as  to  lead  him  to  the 
conclusion  that  they  perform  revolutions,  one  around  the  other, 
in  regular  orbits. 

425.  These  conclusions  have  been  fully  confirmed  by  later  ob- 
servers, so  that  it  is  now  considered  as  fully  established,  that  there 
exist  among  the  fixed  stars,  binary  systems,  in  which  two  stars 
perform  to  each  other  the  office  of  sun  and  planet,  and  that  the 
periods  of  revolution  of  more  than  one  such  pair  have  been  ascer- 
tained with  something  approaching  to  exactness.  Immersions  and 
emersions  of  stars  behind  each  other  have  been  observed,  and  real 
motions  among  them  detected  rapid  enough  to  become  sensible 
and  measurable  in  very  short  intervals  of  time.  The  following 
table  exhibits  the  present  state  of  our  knowledge  on  this  subject. 


Names. 

Period  in  years. 

Major  axis  of  the  orbit. 

Eccentricity. 

v\  Corona?, 
£  Cancri, 
I  Ursae  Majoris, 
70  Ophiuchi, 
Castor, 
(f  Coronae, 
61  Cygni, 
y  Virginis, 
y  Leonis, 

43.40 
55.00 
58.26 
80.34 
252.66 
286.00 
452.00 
628.90* 
1200.00 

7".714 
8.784 
16.172 
7.358 
30.860 
24.000 

0.4164 
0.4667 
0.7582 
0.6112 

0.8335 

From  this  table  it  appears  (1)  that  the  periods  of  the  double  stars 
are  very  various,  ranging  in  the  case  of  those  already  ascertained 
from  forty  three  years  to  one  thousand ;  (2)  that  their  orbits  are 
very  small  ellipses  more  eccentric  than  those  of  the  planets,  the 
greatest  of  which  (that  of  Mercury)  having  an  eccentricity  of  only 
about  .2  of  the  major  axis. 

The  most  remarkable  of  the  binary  stars  is  y  Virginis,  on  account 
not  only  of  the  length  of  its  period,  but  also  of  the  great  diminution 
of  apparent  distance,  and  rapid  increase  of  angular  motion  about 
each  other  of  the  individuals  composing  it.  It  is  a  bright  star 

According  to  E.  P.  Mason,  171  years. 
34 


266  FIXED    STARS. 

of  the  fourth  magnitude,  and  its  component  stars  are  almost  ex- 
actly equal.  It  has  been  known  to  consist  of  two  stars  since  the 
beginning  of  the  eighteenth  century,  their  distance  being  then 
between  six  and  seven  seconds ;  so  that  any  tolerably  good  tele- 
scope would  resolve  it.  Since  that  time,  they  have  been  con- 
stantly approaching,  and  were  in  1838  hardly  more  than  a  single 
second  asunder ;  so  that  no  telescope  that  is  not  of  a  very  supe- 
rior quality,  is  competent  to  show  them  otherwise  than  as  a  single 
star,  somewhat  lengthened  in  one  direction.  It  fortunately  hap- 
pens that  Bradley  (Astronomer  Royal)  in  1718,  noticed,  and  re- 
corded in  the  margin  of  one  of  his  observation  books,  the  apparent 
direction  of  their  line  of  junction,  as  being  parallel  to' that  of  two 
remarkable  stars  a  and  5  of  the  same  constellation,  as  seen  by  the 
naked  eye, — a  remark  which  has  been  of  signal  service  in  the 
investigation  of  their  orbit.  It  is  found  that  it  passed  its  perihelion 
in  June,  1836.  The  period  given  in  the  table  is  that  assigned 
by  Sir  John  Herschel ;  but  later  observations  indicate  a  much 
shorter  period.  In  the  interval  from  1839  to  1841,  the  star  g  Ursae 
Majoris,  completed  a  full  revolution  from  the  epoch  of  the  first 
measurement  of  its  position  in  1781 ;  and  the  regularity  with  which 
it  has  maintained  its  motion,  is  said  to  have  been  exceedingly 
beautiful.* 

426.  The  revolutions  of  the  binary  stars  have  assured  us  of 
that  most  interesting  fact,  that  the  law  of  gravitation  extends  to 
the  fixed  stars.  Before  these  discoveries,  we  could  not  decide 
except  by  a  feeble  analogy  that  this  law  transcended  the  bounds 
of  the  solar  system.  Indeed,  our  belief  of  the  fact  rested  more 
upon  our  idea  of  unity  of  design  in  all  the  works  of  the  Creator, 
than  upon  any  certain  proof;  but  the  revolution  of  one  star  around 
another  in  obedience  to  forces  which  must  be  similar  to  those  that 
govern  the  solar  system,  establishes  the  grand  conclusion,  that  the 
law  of  gravitation  is  truly  the  law  of  the  material  universe. 

We  have  the  same  evidence  (says  Sir  John  Herschel)  of  the 
revolutions  of  the  binary  stars  about  each  other,  that  we  have  of 
those  of  Saturn  and  Uranus  about  the  sun  ;  and  the  correspond* 

*  Ast.  Trans,  v,  35. 


MOTIONS    OF    THE    FIXED    STARS.  267 

ence  between  their  calculated  and  observed  places  in  such  elon- 
gated ellipses,  must  be  admitted  to  carry  with  it  a  proof  of  the 
prevalence  of  the  Newtonian  law  of  gravity  in  their  systems,  of 
the  very  same  nature  and  cogency  as  that  of  the  calculated  and 
observed  places  of  comets  round  the  center  of  our  own  system. 

But  (he  adds)  it  is  not  with  the  revolutions  of  bodies  of  a  plan- 
etary or  cometary  nature  round  a  solar  center  that  we  are  now 
concerned  ;  it  is  with  that  of  sun  around  sun,  each,  perhaps,  ac- 
companied with  its  train  of  planets  and  their  satellites,  closely 
shrouded  from  our  view  by  the  splendor  of  their  respective  suns, 
and  crowded  into  a  space,  bearing  hardly  a  greater  proportion  to 
the  enormous  interval  which  separates  them,  than  the  distances  of 
the  satellites  of  our  planets  from  their  primaries,  bear  to  their  dis- 
tances from  the  sun  itself. 

427.  Some  of  the  fixed  stars  appear  to  have  a  real  motion  in  space. 

The  apparent  change  of  place  in  the  stars  arising  from  the  pre- 
cession of  the  equinoxes,  the  nutation  of  the  earth's  axis,  the  dimi- 
nution of  the  obliquity  of  the  ecliptic,  and  the  aberration  of  light, 
have  been  already  mentioned ;  but  after  all  these  corrections  are 
made,  changes  of  place  still  occur,  which  cannot  result  from  any 
changes  in  the  earth,  but  must  arise  from  changes  in  the  stars 
themselves.  Such  motions  are  called  the  proper  motions  of  the 
stars.  Nearly  2000  years  ago,  Hipparchus  and  Ptolemy  made  the 
most  accurate  determinations  in  their  power  of  the  relative  situa- 
tions of  the  stars,  and  their  observations  have  been  transmitted  to  us 
in  Ptolemy's  Almagest ;  from  which  it  appears  that  the  stars  retain 
at  least  very  nearly  the  same  places  now  as  they  did  at  that  period. 
Still,  the  more  accurate  methods  of  modern  astronomers,  have 
brought  to  light  minute  changes  in  the  places  of  certain  stars 
which  force  upon  us  the  conclusion,  either  that  our  solar  system 
causes  an  apparent  displacement  of  certain  stars,  by  a  motion  of  its 
own  in  space,  or  that  they  have  themselves  a  proper  motion.  Pos- 
sibly, indeed,  both  these  causes  may  operate. 

428.  If  the  sun,  and  of  course  the  earth  which  accompanies 
him,  is  actually  in  motion,  the  fact 'may  become  manifest  from 
the  apparent  approach  of  the  stars  in  the  region  which  he  is  leav- 


268  FIXED    STARS. 

ing,  and  the  recession  of  those  which  lie  in  the  part  of  the  heav- 
ens towards  which  he  is  travelling.  Were  two  groves  of  trees 
situated  on  a  plain  at  some  distance  apart,  and  we  should  go  from 
one  to  the  other,  the  trees  before  us  would  gradually  appear  fur- 
ther and  further  asunder,  while  those  we  left  behind  would  appear 
to  approach  each  other.  Some  years  since,  Sir  William  Herschel 
supposed  he  had  detected  changes  of  this  kind  among  two  sets  of 
stars  in  opposite  points  of  the  heavens,  and  announced  that  the 
solar  system  was  in  motion  towards  a  point  in  the  constellation 
Hercules  ;*  but  other  astronomers  have  not  found  the  changes 
in  question  such  as  would  correspond  to  this  motion,  or  to  any 
motion  of  the  sun  ;  and  while  it  is  a  matter  of  general  belief  that 
the  sun  has  a  motion  in  space,  the  fact  is  not  considered  as  yet 
entirely  proved. 

429.  In  most  cases  where  a  proper  motion  in  certain  stars  has 
been  suspected,  its  annual  amount  has  been  so  small,  that  many 
years  are  required  to  assure  us,  that  the  effect  is  not  owing  to 
some  other  cause  than  a  real  progressive  motion  in  the  stars  them- 
selves ;  but  in  a  few  instances  the  fact  is  too  obvious  to  admit  of 
any  doubt.     Thus  the  two  stars  61  Cygni,  which  are  nearly  equal, 
have  remained  constantly  at  the  same,  or  nearly  at  the  same  dis- 
tance of  15"  for  at  least  fifty  years  past.     Meanwhile  they  have 
shifted  their  local  situation  in  the  heavens,  4'  23",  the  annual 
proper  motion  of  each  star  being  5."8,  by  which  quantity  this 
system  is  every  year  carried  along  in  some  unknown  path,  by  a 
motion  which  for  many  centuries  must  be  regarded  as  uniform 
and  rectilinear.     A  greater  proportion  of  the  double  stars  than 
of  any  other  indicate  proper  motions,  especially  the  binary  stars 
or  those  which  have  a  revolution  around  each   other.     Among 
stars  not  double,  and  no  way  differing  from  the  rest  in  any  other 
obvious  particular,  ^  Cassiopeise  has  the  greatest  proper  motion  of 
any  yet  ascertained,  amounting  to  nearly  4"  annually. 

DISTANCES    OF    THE    FIXED    STARS,  f 

430.  We  cannot  ascertain  the  actual  distance  of  any  of  the  fixed 
*  Phil.  Trans.  1783,  1805,  and  1806.  t  See  Article  V.  of  the  Addenda. 


DISTANCES  OP    THE    FIXED  STARS.  269 

stars,  but  can  certainly  determine  that  the  nearest  star  is  more  than 
(20,000,000,000,000,)  twenty  billions  of  miles  from  the  earth. 

For  all  measurements  relating  to  the  distances  of  the  sun  and 
planets,  the  radius  of  the  earth  furnishes  the  base  line  (Art.  87.) 
The  length  of  this  line  being  known,  and  the  horizontal  parallax 
of  the  body  whose  distance  is  sought,  we  readily  obtain  the  dis- 
tance by  the  solution  of  a  right  angled  triangle.  But  any  star 
viewed  from  the  opposite  sides  of  the  earth,  would  appear  from 
both  stations,  to  occupy  precisely  the  same  situation  in  the  celes- 
tial sphere,  and  of  course  it  would  exhibit  no  horizontal  parallax. 

But  astronomers  have  endeavored  to  find  a  parallax  in  some  of 
the  fixed  stars  by  taking  the  diameter  of  the  earths  orbit  as  a  base 
line.  Yet  even  a  change  of  position  amounting  to  190  millions 
of  miles,  proves  insufficient  to  alter  the  place  of  a  single  star,  from 
which  it  is  concluded  that  the  stars  have  not  even  any  annual  par 
allax ;  that  is,  the  angle  subtended  by  the  semi-diameter  of  the 
earth's  orbit,  at  the  nearest  fixed  star,  is  insensible.  The  errors  to 
which  instrumental  measurements  are  subject,  arising  from  the 
defects  of  the  instruments  themselves,  from  refraction,  and  from 
various  other  sources  of  inaccuracy,  are  such,  that  the  angular 
determinations  of  arcs  of  the  heavens  cannot  be  relied  on  to  less 
than  1".  But  the  change  of  place  in  any  star  when  viewed  at 
opposite  extremities  of  the  earth's  orbit,  is  less  than  1",  and  there- 
fore cannot  be  appreciated  by  direct  measurement.  It  follows, 
that,  when  viewed  from  the  nearest  star,  the  diameter  of  the  earth's 
orbit  would  be  insensible ;  the  spider  line  of  the  telescope  would 
more  than  cover  it. 

431.  Taking,  however,  the  annual  parallax  of  a  fixed  star  at  1", 
let  a  b  (Fig.  73)  represent  the  radius  of  the  earth's  orbit  and  c  a 
fixed  star,  the  angle  at  c  being  1 "  and  the  angle  at  b  a  right  angle ; 
then,, 

Sin.  1"  :  Rad.::l  :  200,000,  nearly. 

Hence  the  hypothenuse  of  a  triangle  whose  vertical  angle  is 
1"  is  about  200,000  times  the  base;  consequently  the  distance  of 
the  nearest  fixed  star  must  exceed  95000000  x  200000=190000000  x 
100000,  or  one  hundred  thousand  times  one  hundred  and  ninety 
millions  of  miles.  Of  a  distance  so  vast  we  can  form  no  adequate 


270  FIXED   STARS. 

conceptions,  and  even  seek  to  measure  it  only  by  the  time 
that  light,  (which  moves  more  than   192,000   miles  per       'c 
second  and  passes  from  the  sun  to  the  earth  in  8m.  13.3 
sec.,)  would  take  to  traverse  it,  which  is  found  to  be  more 
than  three  and  a  half  years. 

If  these  conclusions  are  drawn  with  respect  to  the 
largest  of  the  fixed  stars,  which  we  suppose  to  be  vastly 
nearer  to  us  than  those  of  the  smallest  magnitude,  the  idea 
of  distance  swells  upon  us  when  we  attempt  to  estimate 
the  remoteness  of  the  latter.  As  it  is  uncertain,  however, 
whether  the  difference  in  the  apparent  magnitudes  of  the 
stars  is  owing  to  a  real  difference  or  merely  to  their  being 
at  various  distances  from  the  eye,  more  or  less  uncertainty 
must  attend  all  efforts  to  determine  the  relative  distances 
of  the  stars;  but  astronomers  generally  believe  that  the 
lower  orders  of  stars  are  vastly  more  distant  from  us  than  the 
higher.  Of  some  stars  it  is  said,  that  thousands  of  years  would 
be  required  for  their  light  to  travel  down  to  us. 

432.  We  have  said  that  the  stars  have  no  annual  parallax  ;  yet 
it  may  be  observed  that  astronomers  are  not  exactly  agreed  on 
this  point.     Dr.  Brinkley,  a  late  eminent  Irish  astronomer,  sup- 
posed that  he  had  detected  an  annual  parallax  in  a  Lyrse  amounting 
to  l".13vand  in  a  Aquilae  of  I". 42.*     These  results  were  contro- 
verted by  Mr.  Pond  of  the  Royal  Observatory  of  Greenwich ;  and 
Mr.  Struve  of  Dorpat  has  shown  that  in  a  number  of  cases,  the 
parallax  is  negative,  that  is,  in  a  direction  opposite  to  that  which 
would  arise  from  the  motion  of  the  earth.     Hence,  until  recently, 
it  was  considered  doubtful  whether  in  all  cases  of  an  apparent 
parallax,  the  effect  is  not  wholly  due  to  errors  of  observation. 

433.  Indirect  methods  have  been  proposed  for  ascertaining  the 
parallax  of  the  fixed  stars  by  means  of  observations  on  the  double 
stars.     If  the  two  stars  composing  a  double  star  are  at  different 
distances  from  us,  parallax  would  affect  them  unequally,  and  change 
their  relative  position  with  respect  to  each  other ;  and  since  the 

*  Phil.  Trans.  1821. 


NATURE   OF   THE   FIXED    STARS.  271 

ordinary  sources  of  error  arising  from  the  imperfection  of  instru- 
ments, from  precession,  nutation,  aberration,  and  refraction,  would 
be  avoided,  (as  they  would  affect  both  objects  alike,  and  there- 
fore would  not  disturb  their  relative  positions,)  measurements 
taken  with  the  micrometer  of  changes  much  less  than  1"  may  be 
relied  on.  Sir  John  Herschel  proposes  a  method*  by  which 
changes  may  be  determined  which  amount  to  only  +\  of  a  second.f 

434.  The  immense  distance  of  the  fixed  stars  is  inferred  also 
from  the  fact  that  the  largest  telescopes  do  not  increase  their  ap- 
parent magnitude.     They  are  still  points,  when  viewed  with  the 
highest  magnifiers,  although  they  sometimes  present  a  spurious 
disk,  which  is  owing  to  irradiation.} 

NATURE    OF    THE    STARS. 

435.  The  stars  are  bodies .  greater  than  our  earth.     If  this  were 
not  the  case  they  could  not  be  visible  at  such  an  immense  distance. 
Dr.  Wollaston,  a  distinguished  English  philosopher,  attempted  to  esti- 
mate the  magnitudes  of  certain  of  the  fixed  stars  from  the  light  which 
they  afford.     By  means  of  an  accurate  photometer  (an  instrument 
for  measuring  the  relative  intensities  of  light)  he  compared  the 
light  of  Sirius  with  that  of  the* sun.     He  next  inquired  how  far  the 
sun  must  be  removed  from  us  in  order  to  appear  no  brighter  than 
Sirius.     He  found  the  distance  to  be  141,400  times  its  present  dis- 
tance.    But  Sirius  is  more  than  200,000  times  as  far  off  as  the  sun 
(Art.  431.)     Hence  he  inferred  that,  upon  the  lowest  computation, 
Sirius  must  actually  give  out  twice  as  much  light  as  the  sun ;  or 
that,  in  point  of  splendor,  Sirius  must  be  at  least  equal  to  two 

»  Phil.  Trans.  1826. 

t  Very  recent  intelligence  informs  us,  that  Professor  Bessel  of  Konigsberg,  has  ob- 
tained decisive  evidence  of  an  annual  parallax  in  61  Cygni,  amounting  to  0."3483. 
This  makes  the  distance  of  that  star,  equal  to  592000  times  95  millions  of  miles, — a 
distance  which  it  would  take  light  9£  years  to  traverse. 

t  Irradiation  is  an  enlargement  of  objects  beyond  their  proper  bounds,  in  consequence 
of  the  vivid  impression  of  light  on  the  eye.  It  is  supposed  to  increase  the  apparent  di- 
ameters of  the  sun  and  moon  from  three  to  four  seconds,  and  to  create  an  appearance 
of  a  disk  in  a  fixed  star  which,  when  this  cause  is  removed,  is  seen  as  a  mere  point.  See 
Richardson,  Astr.  Trans,  v,  1. 


272  FIXED    STARS. 

suns.     Indeed,  he  has  rendered  it  probable  that  the  light  of  Sinus 
is  equal  to  fourteen  suns. 

436.  The  fixed  stars  are  suns.     We  have  already  seen  that  they 
are  large  bodies;  that  they  are  immensely  further  off  than  the 
furthest  planet ;  that  they  shine  by  their  own  light :  in  short,  that 
their  appearance  is,  in  all  respects,  the  same  as  the  sun  would  ex- 
hibit if  removed  to  the  region  of  the  stars.     Hence  we  infer  that 
they  are  bodies  of  the  same  kind  with  the  sun. 

437.  We  are  justified  therefore  by  a  sound  analogy,  in  concluding 
that  the  stars  were  made  for  the  same  end  as  the  sun,  namely,  as 
the  centers  of  attraction  to  other  planetary  worlds,  to  which  they 
severally  dispense  light  and  heat.     Although  the  starry  heavens 
present,  in  a  clear  night,  a  spectacle  of  ineffable  grandeur  and 
beauty,  yet  it  must  be  admitted  that  the  chief  purpose  of  the  stars 
could  not  have  been  to  adorn  the  night,  since  by  far  the  greatest 
part  of  them  are  wholly  invisible  to  the  naked  eye ;  nor  as  land- 
marks to  the  navigator,  for  only  a  very  small  proportion  of  them 
are  adapted  to  this  purpose ;  nor,  finally,  to  influence  the  earth  by 
their  attractions,  since  their  distance  renders  such  an  effect  entirely 
insensible.    If  they  are  suns,  and  if  they  exert  no  important  agen- 
cies upon  our  world,  but  are  bodies  evidently  adapted  to  the  same 
purpose  as  our  sun,  then  it  is  as  rational  to  suppose  that  they  were 
made  to  give  light  and  heat,  as  that  the  eye  was  made  for  seeing 
and  the  ear  for  hearing.     It  is  obvious  to  inquire  next,  to  what 
they  dispense  these  gifts  if  not  to  planetary  worlds  ;  and  why  to 
planetary  worlds,  if  not  for  the  use  of  percipient  beings  ?     We 
are    thus   led,  almost  inevitably,  to  the  idea  of  a  Plurality  of 
Worlds ;  and  the  conclusion  is  forced  upon  us,  that  the  spot  which 
the  Creator  has  assigned  to  us  is  but  a  humble  province  of  his 
boundless  empire.* 

*  See  this  argument,  in  its  full  extent,  in  Dick's  Celestial  Scenery. 


CHAPTER  IV. 

OP   THE    SYSTEM    OF   THE    WORLD. 

438.  The  arrangement  of  all  the  bodies  that  compose  the  material 
universe,  and  their  relations  to  each  other,  constitute  the  System  of 
the  World. 

It  is  otherwise  called  the  Mechanism  of  the  Heavens ;  and  in- 
deed in  the  System  of  the  world,  we  figure  to  ourselves  a  machine, 
all  the  parts  of  which  have  a  mutual  dependence,  and  conspire  to 
one  great  end.  "  The  machines  that  are  first  invented  (says  Adam 
Smith)  to  perform  any  particular  movement,  are  always  the  most 
complex  ;  and  succeeding  artists  generally  discover  that  with  fewer 
wheels  and  with  fewer  principles  of  motion  than  had  originally 
been  employed,  the  same  effects  may  be  more  easily  produced* 
The  first  systems,  in  the  same  manner,  are  always  the  most  com- 
plex ;  and  a  particular  connecting  chain  or  principle  is  generally 
thought  necessary  to  unite  every  two  seemingly  disjointed  appear- 
ances ;  but  it  often  happens,  that  one  great  connecting  principle  is 
afterwards  found  to  be  sufficient,  to  bind  together  all  the  discord- 
ant phenomena  that  occur  in  a  whole  species  of  things."  This  re- 
mark is  strikingly  applicable  to  the  origin  and  progress  of  systems 
of  astronomy. 

439.  From  the  visionary  notions  which  are  generally  understood 
to  have  been  entertained  on  this  subject  by  the  ancients,  we  are 
apt  to  imagine  that  they  knew  less  than  they  actually  did  of  the 
truths  of  astronomy.     But  Pythagoras,  who  lived  500  years  before 
the  Christian  era,  was  acquainted  with  many  important  facts  in 
our  science,  and  entertained  many  opinions  respecting  the  system 
of  the  world  which  are  now  held  to  be  true.     Among  other  things 
well  known  to  Pythagoras  were  the  following  : 

1.  The  principal  Constellations.  These  had  begun  to  be  formed 
in  the  earliest  ages  of  the  world.  Several  of  them  bearing  the 
same  names  as  at  present  are  mentioned  in  the  writings  of  Hesiod 

35 


274  SYSTEM    OP   THE    WORLD. 

and  Homer  ;  and  the  "  sweet  influences  of  the  Pleiades"  and  the 
"  bands  of  Orion,"  are  beautifully  alluded  to  in  the  book  of  Job. 

2.  Eclipses.     Pythagoras  knew  both  the  causes  of  eclipses  and 
how  to  predict  them  ;*  not  indeed  in  the  accurate  manner  now 
employed,  but  by  means  of  the  Saros  (Art.  233.) 

3.  Pythagoras  had  divined  the  true  system  of  the  world,  hold- 
ing that  the  sun  and  not  the  earth,  (as  was  generally  held  by  the 
ancients,  even  for  many  ages   after   Pythagoras,)    is  the  center 
around  which  all  the  planets  revolve,  and  that  the  stars  are  so  many 
suns,  each  the  center  of  a  system  like  our  own.f     Among  lesser 
things,  he  knew  that  the  earth  is  round  ;  that  its  surface  is  naturally 
divided  into  five  Zones ;  and  that  the  ecliptic  is  inclined  to  the 
equator.     He  also  held  that  the  earth  revolves  daily  on  its  axis,  and 
yearly  around  the  sun ;  that  the  galaxy  is  an  assemblage  of  small 
stars  ;  and  that  it  is  the  same  luminary,  namely,  Venus,  that  con- 
stitutes both  the  morning  and  the  evening  star,  whereas  all  the  an- 
cients before  him  had  supposed  that  each  was  a  separate  planet, 
and  accordingly   the  morning  star   wras  called  Lucifer,  and  the 
evening  star  Hesperus.J     He  held  also  that  the  planets  were  in- 
habited, and  even  went  so  far  as  to  calculate  the  size  of  some  of 
the  animals  in  the  moon.§     Pythagoras  was  so  great  an  enthusiast 
in  music,  that  he  not  only  assigned  to  it  a  conspicuous  place  in  his 
system  of  education,  but  even  supposed  the  heavenly  bodies  them- 
selves to  be  arranged  at  distances  corresponding  to  the  diatonic 
scale,  and  imagined  them  to  pursue  their  sublime  march  to  notes 
created  by  their  own  harmonious  movements,  called  the  "  music 
of  the  spheres ;"   but  he  maintained   that   this  celestial  concert, 
though  loud  and  grand,  is  not  audible  to  the  feeble  organs  of  man, 
but  only  to  the  gods. 

440.  With  few  exceptions,  however,  the  opinions  of  Pythago- 
ras on  the  System  of  the  World,  were  founded  in  truth.  Yet  they 
were  rejected  by  Aristotle  and  by  most  succeeding  astronomers 
down  to  the  time  of  Copernicus,  and  in  their  place  was  substituted 


*  Long's  Astronomy,  2.  671. 

t  Library  of  Useful  Knowledge,  History  of  Astronomy. 

t  Long'a  Ast.  2.  673.  §  Ed.  Encyclopaedia. 


ASTRONOMICAL    KNOWLEDGE    OF   THE    ANCIENTS.  275 

the  doctrine  of  Cyrstalline  Spheres,  first  taught  by  Eudoxus.  Ac- 
cording to  this  system,  the  heavenly  bodies  are  set  like  gems  in 
hollow  solid  orbs,  composed  of  crystal  so  pellucid  that  no  anterior 
orb  obstructs  in  the  least  the  view  of  any  of  the  orbs  that  lie  behind 
it.  The  sun  and  the  planets  have  each  its  separate  orb  ;  but  the 
fixed  stars  are  all  set  in  the  same  grand  orb ;  and  beyond  this  is 
another  still,  the  Primum  Mobile,  which  revolves  daily  from  east 
to  west,  and  carries  along  with  it  all  the  other  orbs.  Above  the 
whole,  spreads  the  Grand  Empyrean,  or  third  heavens,  the  abode 
of  perpetual  serenity.* 

To  account  for  the  planetary  motions,  it  was  supposed  that  each 
of  the  planetary  orbs  as  well  as  that  of  the  sun,  has  a  motion  of  its 
own  eastward,  while  it  partakes  of  the  common  diurnal  motion  of 
the  starry  sphere.  Aristotle  taught  that  these  motions  are  effected 
by  a  tutelary  genius  of  each  planet,  residing  in  it,  and  directing  its 
motions,  as  the  mind  of  man  directs  his  motions. 

441.  On  coming  down  to  the  time  of  Hipparchus,  who  flourished 
about  150  years  before  the  Christian  era,*we  meet  with  astrono- 
mers who  acquired  far  more  accurate  knowledge  of  the  celestial 
motions.  Hipparchus  was  in  possession  of  instruments  for  meas- 
uring angles,  and  knew  how  to  resolve  spherical  triangles.  He 
ascertained  the  length  of  the  year  within  6m.  of  the  truth.  He 
discovered  the  eccentricity  of  the  solar  orbit,  (although  he  supposed 
the  sun  actually  to  move  uniformly  in  a  circle,  but  the  earth  to  be 
placed  out  of  the  center,)  and  the  positions  of  the  sun's  apogee  and 
perigee.  He  formed  very  accurate  estimates  of  the  obliquity  of 
the  ecliptic  and  of  the  precession  of  the  equinoxes.  He  computed 
the  exact  period  of  the  synodic  revolution  of  the  moon,  and  the 
inclination  of  the  lunar  orbit ;  discovered  the  motion  of  her  node 
and  of  her  line  of  apsides  ;  and  made  the  first  attempts  to  ascer- 
tain the  horizontal  parallaxes  of  the  sun  and  moon. 

Such  was  the  state  of  astronomical  knowledge  when  Ptolemy 
wrote  the  Almagest,  in  which  he  has  transmitted  to  us  an  ency- 
clopaedia of  the  astronomy  of  the  ancients. 


*  Long's  Ast.  2.  640— Robinson's  Mcch.  Phil.  2.  83— Gregory's  Ast,  132— Play  fair's 
Dissertation,  118. 


276  SYSTEM    OF    THE    WORLD. 

442.  The  systems  of  the  world  which  have  been  most  celebrated 
are  three — the   Ptolemaic,  the  Tychonic,   and  the   Copernican. 
We  shall  conclude  this  part  of  our  work  with  a  concise  statement 
and  discussion  of  each  of  these  systems  of  the  Mechanism  of  the 
Heavens. 

THE    PTOLEMAIC    SYSTEM. 

443.  The  doctrines  of  the  Ptolemaic  System  were  not  originated 
by  Ptolemy,  but  being  digested  by  him  out  of  materials  furnished 
by  various  hands,  it  has  come  down  to  us  under  the  sanction  of 
his  name. 

According  to  this  system,  the  earth  is  the  center  of  the  universe, 
and  all  the  heavenly  bodies  daily  revolve  around  it  from  east  to 
west.  In  order  to  explain  the  planetary  motions,  Ptolemy  had  re- 
course to  deferents  and  epicycles, — an  explanation  devised  by  Apol- 
lonius,  one  of  the  greatest  geometers  of  antiquity.*  He  conceived 
that,  in  the  circumference  of  a  circle,  having  the  earth  for  its  cen- 
ter, there  moves  the  center  of  another  circle,  in  the  circumference 
of  which  the  planet  actually  revolves.  The  circle  surrounding  the 
earth  was  called  the  deferent,  while  the  smaller  circle,  whose  center 
was  always  in  the  periphery  of  the  deferent,  was  called  the  epi- 
cycle. The  motion  in  each  was  supposed  to  be  uniform.  Lastly, 
it  was  conceived  that  the  motion  of  the  center  of  the  epicycle  in 
the  circumference  of  the  deferent,  and  of  tlie  deferent  itself,  are 
in  opposite  directions,  the  first  being  towards  the  east,  and  the 
second  towards  the  west. 

444.  But  these  views  will  be  better  understood  from  a  diagram. 
Therefore,  let  ABC  (Fig.  74,)  represent  the  deferent,  E  being  the 
earth  a  little  out  of  the  center.     Let  abc  represent  the  epicycle, 
having  its  center  at  v,  on  the  periphery  of  the  deferent.     Con- 
ceive the  circumference  of  the  deferent  to  be  carried  about  the 
earth  every  twenty-four  hours  in  the  order  of  the  letters ;  and  at 
the  same  time,  let  the  center  v  of  the  epicycle  abed,  have  a  slow 
motion  in  the  opposite  direction,  and  let  a  body  revolve  in  this  cir- 

*  Playfair,  Dissertation  Second,  119. 


THE    PTOLEMAIC    SYSTEM.  277 

cle  in  the  direction  alcd.     Then  it  will  be  seen  that  the  body 
would  actually  describe  the  looped  curves  klmnop ;  that  it  would 

(Fig.  74.) 


appear  stationary  at  Z  and  m,  and  at  n  and  o;  that  its  motion 
would  be  direct  from  7c  to  /,  and  then  retrograde  from  I  to  m ;  di- 
rect again  from  m  to  n,  and  retrograde  from  n  to  o. 

445.  Such  a  deferent  and  epicycle  may  be  devised  for  each 
planet  as  will  fully  explain  all  its  ordinary  motions ;  but  it  is  in- 
consistent with  the  phases  of  Mercury  and  Venus,  which  being  be- 
tween us  and  the  sun  on  both  sides  of  the  epicycle,  would  present 
their  dark  sides  towards  us  in  both  these  positions,  whereas  at  one 
of  the  conjunctions  they  are  seen  to  shine  with  full  face.*  It  is 
moreover  absurd  to  speak  of  a  geometrical  center,  which  has  no 
bodily  existence,  moving  around  the  earth  on  the  circumference 
of  another  circle ;  and  hence  some  suppose  that  the  ancients 
merely  assumed  this  hypothesis  as  affording  a  convenient  geome- 
trical representation  of  the  phenomena, — a  diagram  simply,  with- 
out conceiving  the  system  to  have  any  real  existence  in  nature. 

*  Vince's  Complete  System,  I,  96. 


278  SYSTEM    OF    THE    WORLD. 

446.  The  objections  to  the  Ptolemaic  system,  in  general,  are  the 
following :  First,  it  is  a  mere  hypothesis,  having  no  evidence  in  its 
favor,  except  that  it  explains  the  phenomena.  This  evidence  is 
insufficient  of  itself,  since  it  frequently  happens  that  each  of  two 
hypotheses,  directly  opposite  to  each  other,  will  explain  all  the 
known  phenomena.  But  the- Ptolemaic  system  does  not  even  do 
this,  as  it  is  inconsistent  with  the  phases  of  Mercury  and  Venus, 
as  already  observed.  Secondly,  now  that  we  are  acquainted  with 
the  distances  of  the  remoter  planets,  and  especially  of  the  fixed 
stars,  the  swiftness  of  motion  implied  in  a  daily  revolution  of  the 
starry  firmament  around  the  earth,  renders  such  a  motion  wholly 
incredible.  Thirdly,  the  centrifugal  force  that  would  be  generated 
in  these  bodies,  especially  in  the  sun,  renders  it  impossible  that 
they  can  continue  to  revolve  around  the  earth  as  a  center. 

These  reasons  are  sufficient  to  show  the  absurdities  of  the 
Ptolemaic  System  of  the  World. . 


THE   TYCHONIC    SYSTEM. 

447.  Tycho  Brahe,  like  Ptolemy,  placed  the  earth  in  the  center 
of  the  universe,  and  accounted  for  the  diurnal  motions  in  the  same 
manner  as  Ptolemy  had  done,  namely,  by  an  actual  revolution  of 
the  whole  host  of  heaven  around  the  earth  every  twenty-four 
hours.     But  he  rejected  the  scheme  of  deferents  and  epicycles,  and 
held  that  the  moon  revolves  about  the  earth  as  the  center  of  her 
motions  ;  that  the  sun,  and  not  the  earth,  is  the  center  of  the  plan- 
etary motions ;    and  that  the  sun  accompanied  by   the   planets 
moves  around  the  earth  once  a  year,  somewhat  in  the  manner  that 
we  now  conceive  of  Jupiter  and  his  satellites  as  revolving  around 
the  sun. 

448.  The  system  of  Tycho  serves  to  explain  all  the  common 
phenomena  of  the  planetary  motions,  but  it  is  encumbered  with 
the  same  objections  as  those  that  have  been  mentioned  as  resting 
against  the  Ptolemaic  system,  namely,  that  it  is  a  mere  hypothesis ; 
that  it  implies  an  incredible  swiftness  in  the  diurnal  motions  ;  and 
that  it  is  inconsistent  with  the  known  laws  of  universal  gravitation. 


THE    COPERNICAN    SYSTEM.  279 

But  if  the  heavens  do  not  revolve,  the  earth  must,  and  this  brings 
us  to  the  system  of  Copernicus. 


THE    COPERNICAN    SYSTEM. 

449.  Copernicus  was  born  at  Thorn  in  Prussia  in  1473.     The 
system  that  bears  his  name  was  the  fruit  of  forty  years  of  intense 
study  and  meditation  upon  the  celestial  motions.     As  already  men- 
tioned, (Art.  6,)  it  maintains  (1)  That  the  apparent  diurnal  motions 
of  the  heavenly  bodies,  from  east  to  west,  is  owing  to  the  real 
revolution  of  the  earth  on  its  own  axis  from  west  to  east ;  and  (2) 
That  the  sun  is  the  center  around  which  the  earth  and  planets  all 
revolve  from  west  to  east.     It  rests  on  the  following  arguments : 

450.  First,  the  earth  revolves  on  its  own  axis. 

1.  Because  this  supposition  is  vastly  more  simple. 

2.  It  is  agreeable  to  analogy,  since  all  the  other  planets  that 
afford  any  means  of  determining  the  question,  are  seen  to  revolve 
on  their  axes. 

3.  The  spheroidal  figure  of  the  earth,  is  the  figure  of  equilibrium, 
that  results  from  a  revolution  on  its  axis. 

4.  The  diminished  weight  of  bodies  at  the  equator,  indicates  a 
centrifugal  force  arising  from  such  a  revolution. 

5.  Bodies  let  fall  from  a  high  eminence,  fall  eastward  of  their 
base,  indicating  that  when  further  from  the  center  of  the  earth 
they  were  subject  to  a  greater  velocity,  which  in  consequence  of 
their  inertia,  they  do  not  entirely  lose  in  descending  to  the  lower 
level* 

451.  Secondly,  the  planets,  including  the  earth,  revolve  about  the 
sun. 

1.  The  phases  of  Mercury  and  Venus  are  precisely  such,  as  would 
result  from  their  circulating  around  the  sun  in  orbits  within  that 
of  the  earth  ;  but  they  are  never  seen  in  opposition,  as  they  would 
be  if  they  circulated  around  the  earth. 

2.  The  superior  planets  do  indeed  revolve  around  the  earth ; 
but  they  also  revolve  around  the  sun,  as  is  evident  from  their  phases 

*  Biot. 


280  SYSTEM    OF    THE    WORLD. 

and  from  the  known  dimensions  of  their  orbits ;  and  that  the  sun 
and  not  the  earth,  is  the  center  of  their  motions,  is  inferred  from 
the  greater  symmetry  of  their  motions  as  referred  to  the  sun  than 
as  referred  to  the  earth,  and  especially  from  the  laws  of  gravitation, 
which  forbid  our  supposing  that  bodies  so  much  larger  than  the 
earth,  as  some  of  these  bodies  are,  can  circulate  permanently  around 
the  earth,  the  latter  remaining  all  the  while  at  rest. 

3.  The  annual  motion  of  the  earth  itself  is  indicated  also  by  the 
most  conclusive  arguments.  For,  first,  since  all  the  planets  with 
their  satellites,  and  the  comets,  revolve  about  the  sun,  analogy 
leads  us  to  infer  the  same  respecting  the  earth  and  its  satellite. 
Secondly,  the  motions  of  the  satellites,  as  those  of  Jupiter  and 
Saturn,  indicate  that  it  is  a  law  of  the  solar  system  that  the  smaller 
bodies  revolve  about  the  larger.  Thirdly,  on  the  supposition  that 
the  earth  performs  an  annual  revolution  around  the  sun.  it  is  em- 
braced along  with  the  planets,  in  Kepler's  law,  that  the  squares  of 
the  times  are  as  the  cubes  of  the  distances ;  otherwise,  it  forms  an 
exception,  and  the  only  known  exception  to  this  law.  Lastly,  the 
aberration  of  light  affords  a  sensible  proof  of  the  motion  of  the 
earth,  since  that  phenomenon  indicates  both  a  progressive  motion 
of  light,  and  a  motion  of  the  earth  from  west  to  east.  (Art.  195.) 

452.  It  only  remains  to  inquire,  whether  there  subsist  higher 
orders  of  relations  between  the  stars  themselves. 

The  revolutions  of  the  binary  stars  (Art.  424)  afford  conclusive 
evidence  of  at 'least  subordinate  systems  of  suns,  governed  by  the 
the  same  laws  as  those  which  regulate  the  motions  of  the  solar 
system.  The  nebulce  also  compose  peculiar  systems,  in  which  the 
members  are  evidently  bound  together  by  some  common  relation. 

In  these  marks  of  organization, — of  stars  associated  together  in 
clusters, — of  sun  revolving  around  sun,  and  of  nebulae  disposed 
in  regular  figures,  we  recognize  different  members  of  some  grand 
system,  links  in  one  great  chain  that  binds  together  all  parts  of 
the  universe  ;  as  we  see  Jupiter  and  his  satellites  combined  in  one 
subordinate  system,  and  Saturn  and  his  satellites  in  another, — each 
a  vast  kingdom,  and  both  uniting  with  a  number  of  other  indi- 
vidual parts  to  compose  an  empire  still  more  vast. 


THE    COPERNICAN    SYSTEM.  281 

453.  This  fact  being  now  established,  that  the  stars  are  immense 
bodies  like  the  sun,  and  that  they  are  subject  to  the  laws  of  gravi- 
tation, we  cannot  conceive  how  they  can  be  preserved  from  falling 
into  final  disorder  and  ruin,  unless  they  move  in  harmonious  con- 
cert like  the  members  of  the  solar  system.  Otherwise,  those  that 
are  situated  on  the  confines  of  creation,  being  retained  by  no  forces 
from  without,  while  they  are  subject  to  the  attraction  of  all  the 
bodies  within,  must  leave  their  stations,  and  move  inward  with 
accelerated  velocity,  and  thus  all  the  bodies  in  the  universe  would 
at  length  fall  together  in  the  common  center  of  gravity.  The 
immense  distance  at  which  the  stars  are  placed  from  each  other, 
would  indeed  delay  such  a  catastrophe  ;  but  such  must  be  the  ulti- 
mate tendency  of  the  material  world,  unless  sustained  in  one  har- 
monious system  by  nicely  adjusted  motions.*  To  leave  entirely 
out  of  view  our  confidence  in  the  wisdom  and  preserving  goodness 
of  the  Creator,  and  reasoning  merely  from  what  we  know  of  the 
stability  of  the  solar  system,  we  should  be  justified  in  inferring, 
that  other  worlds  are  not  subject  to  forces  which  operate  only  to 
hasten  their  decay,  and  to  involve  them  in  final  ruin. 

We  conclude,  therefore,  that  the  material  universe  is  one  great 
system ;  that  the  combination  of  planets  with  their  satellites  con- 
stitutes the  first  or  lowest  order  of  worlds ;  that  next  to  these 
planets  are  linked  to  suns ;  that  these  are  bound  to  other  suns, 
composing  a  still  higher  order  in  the  scale  of  being ;  and,  finally, 
that  all  the  different  systems  of  worlds,  move  around  their  common 
center  of  gravity,  f 

*  Robison's  Physical  Astronomy.  t  See  Article  V.  of  the  Addenda. 

36 


ADDENDA. 

The  very  liberal  patronage  which  this  work  has  received,  enables  us 
to  add  a  few  articles  of  interest  and  importance  not  introduced  into  the 
earlier  editions. 


ARTICLE  I. 

METEORIC    SHOWERS. 

THE  remarkable  exhibitions  of  shooting  stars  which  have  occurred 
within  a  few  years  past,  have  excited  great  interest  among  astronomers, 
and  led  to  some  new  views  respecting  the  construction  of  the  solar  sys- 
tem. Their  attention  was  first  turned  towards  this  subject  by  the  great 
meteoric  shower  of  November  13th,  1833.  On  that  morning,  from  two 
o'clock  until  broad  daylight,  the  sky  being  perfectly  serene  and  cloudless, 
the  whole  heavens  were  lighted  with  a  magnificent  display  of  celestial 
fireworks.  At  times,  the  air  was  filled  with  streaks  of  light,  occasioned 
by  fiery  particles  darting  down  so  swiftly  as  to  leave  the  impression  of 
their  light  on  the  eye,  (like  a  match  ignited  and  whirled  before  the  face,) 
and  drifting  to  the  northwest  like  flakes  of  snow  driven  by  the  wind  ; 
while,  at  short  intervals,  balls  of  fire,  varying  in  size  from  minute  points 
to  bodies  larger  than  Jupiter  and  Venus,  and  in  a  few  instances  as  large 
as  the  full  moon,  descended  more  slowly  along  the  arch  of  the  sky,  often 
leaving  after  them  long  trains  of  light,  which  were,  in  some  instances, 
variegated  with  different  prismatic  colors. 

On  tracing  back  the  lines  of  direction  in  which  the  meteors  moved,  it 
was  found  that  they  all  appeared  to  radiate  from  the  same  point,  which 
was  situated  near  one  of  the  stars  (Gamma  Leonis)  of  the  sickle,  in  the 
constellation  Leo ;  and,  in  every  repetition  of  the  meteoric  shower  of  No- 
vember,  the  radiant  point  has  occupied  nearly  the  same  situation. 

This  shower  pervaded  nearly  the  whole  of  North  America,  having  ap- 
peared in  almost  equal  splendor  from  the  British  possessions  on  the  north, 
to  the  West  India  islands  and  Mexico  on  the  south,  and  from  sixty-one  de- 
grees of  longitude  east  of  the  American  coast,  quite  to  the  Pacific  ocean 
on  the  west.  Throughout  this  immense  region,  the  duration  was  nearly 
the  same.  The  meteors  began  to  attract  attention  by  their  unusual  fre- 
quency and  brilliancy,  from  nine  to  twelve  o'clock  in  the  evening  ;  were 
most  striking  in  their  appearance  from  two  to  four  ;  arrived  at  their  maxi- 
mum, in  many  places,  about  four  o'clock  ;  and  continued  until  rendered 
invisible  by  the  light  of  day.  The  meteors  moved  in  right  lines,  or  in 


ADDENDA.  283 

such  apparent  curves,  as,  upon  optical  principles,  can  be  resolved  into 
right  lines.  Their  general  tendency  was  towards  the  northwest,  although 
by  the  effect  of  perspective  they  appeared  to  move  in  all  directions. 

Soon  after  this  occurrence,  it  was  ascertained  that  a  similar  meteoric 
shower  had  appeared  in  1799,  and  what  was  remarkable,  almost  exactly 
at  the  same  time  of  year,  namely,  on  the  morning  of  the  12th  of  Novem- 
ber; and  it  soon  appeared,  by  accounts  received  from  different  parts  of 
the  world,  that  this  phenomenon  had  occurred  on  the  same  13th  of  No- 
vember, in  1830,  1831,  and  1832.  Hence,  this  was  evidently  an  event 
independent  of  the  casual  changes  of  the  atmosphere  ;  for,  having  a  pe- 
riodical return,  it  was  undoubtedly  to  be  referred  to  astronomical  causes, 
and  its  recurrence,  at  a  certain  definite  period  of  the  year,  plainly  indi- 
cated some  relation  to  the  revolution  of  the  earth  around  the  sun. 

It  remained,  however,  to  develop  the  nature  of  this  relation,  by  inves- 
tigating, if  possible,  the  origin  of  the  meteors.  The  views  to  which  the 
author  of  this  work  was  led,  suggested  the  probability  that  the  same  phe- 
nomenon would  recur  at  the  corresponding  seasons  of  the  year  for  at  least 
several  years  afterwards  ;  and  such  proved  to  be  the  fact,  although  the 
appearances,  at  every  succeeding  return,  were  less  and  less  striking,  until 
1839,  when,  so  far  as  is  known,  they  ceased  altogether. 

Meanwhile,  three  other  distinct  periods  of  meteoric  showers  have  also 
been  determined  ;  one  about  the  21st  of  April,  another  on  the  9th  of  Au- 
gust, and  another  on  the  7th  of  December. 

The  following  conclusions  respecting  the  meteoric  shower  of  November, 
are  believed  to  be  well  established,  and  most  of  them  are  now  generally 
admitted  by  astronomers,  though  we  cannot  here  exhibit  the  evidence  on 
which  they  were  founded,  but  must  beg  leave  to  refer  the  reader  to  vari- 
ous publications  on  this  subject  in  the  American  Journal  of  Science,  com- 
mencing with  the  25th  volume  ;  and  also  to  "  Letters  on  Astronomy,"  by 
the  author  of  this  work. 

It  is  considered,  then,  as  established,  that  the  meteors  had  their  origin 
beyond  the  limits  of  the  atmosphere,  having  descended  to  us  from  some 
body  existing  in  space  independent  of  the  earth  ;  that  they  consisted  of 
exceedingly  light  combustible  matter ;  that  they  moved  with  very  great 
velocities,  amounting  in  some  instances  to  not  less  than  fourteen  miles 
per  second  ;  that  some  of  them  were  bodies  of  large  size,  probably  seve- 
ral hundred  fe'et  in  diameter ;  that  when  they  entered  the  atmosphere, 
they  rapidly  and  powerfully  condensed  the  air  before  them,  and  thus 
elicited  the  heat  which  set  them  on  fire,  as  a  spark  is  sometimes  evolved 
by  condensing  air  suddenly  by  a  piston  and  cylinder ;  and  that  they  were 
consumed  and  resolved  into  small  clouds  at  the  height  of  about  thirty 
miles  above  the  earth. 


284  ADDENDA. 

Calling  the  body  from  which  the  meteors  descended  the  "  meteoric 
body,"  it  is  inferred  that  it  is  a  body  of  great  extent,  since,  without  appa- 
rent exhaustion,  it  has  been  able  to  afford  such  copious  showers  of  mete- 
ors at  so  many  different  times  ;  and  hence  we  regard  the  part  that  has 
descended  to  the  earth  only  as  the  extreme  portions  of  a  body  or  collection 
of  meteors,  of  unknown  extent,  existing  in  the  planetary  spaces.  Since 
the  earth  fell  in  with  the  meteoric  body,  in  the  same  part  of  its  orbit  for 
several  years  in  succession,  the  body  must  either  have  remained  there 
while  the  earth  was  performing  its  whole  revolution  around  the  sun,  or  it 
must  itself  have  had  a  revolution,  as  well  as  the  earth.  No  body  can 
remain  stationary  within  the  planetary  spaces ;  for,  unless  attracted  to 
some  nearer  body,  it  would  be  drawn  directly  towards  the  sun,  and 
could  not  have  been  encountered  by  the  earth  again  in  the  same  part  of 
her  orbit.  Nor  can  any  mode  be  conceived  in  which  this  event  could 
have  happened  so  many  times  in  regular  succession,  unless  the  body  had 
a  revolution  of  its  own  around  the  sun.  Finally,  to  have  come  into  con- 
tact with  the  earth  at  the  same  part  of  her  orbit,  in  two  or  more  successive 
years,  the  body  must  have  either  a  period  which  is  nearly  the  same  with 
the  earth's  period,  or  some  aliquot  part  of  it.  No  period  will  fulfil  the 
conditions,  but  either  a  year  or  half  a  year.  Which  of  these  is  the  true 
period  of  the  meteoric  body,  is  not  fully  determined. 


ARTICLE  II. 

FOUR  GREAT  MECHANICAL  PRINCIPLES. 

(From  Pont^coulant's  Trait6  Etementaire  de  Physique  Celeste,  p.  503.) 

"  Mathematical  analysis  has  furnished  to  geometers  all  the  resources 
which  were  necessary  in  order  to  rise  from  the  consideration  of  the  laws 
of  motion  of  a  single  material  point,  to  that  of  the  motions  of  a  system  of 
bodies  connected  together  in  any  manner  whatsoever.  It  would  be  dif- 
ficult, without  the  help  of  the  calculus,  to  give  even  an  idea  of  the  means 
which  have  been  employed  for  their  discovery ;  but  there  have  been  de- 
duced from  analytical  formulas  several  general  principles,  which  mani- 
fest themselves  in  the  motions  of  every  system  of  bodies,  and  which  it  is 
important  to  make  known,  since  they  have  an  immediate  application  to 
the  constitution  of  our  planetary  system.  These  general  principles  of 
motion  are  four  in  number,  and  may  be  enumerated  as  follows : 

"1.  If  a  system  of  bodies  act  on  one  another  in  any  manner  whatso. 


ADDENDA.  285 

ever,  and  are  subject  to  the  action  of  a  force  directed  towards  a  fixed  cen- 
ter, and  are  not  acted  on  by  any  extraneous  force,  the  sum  of  the  areas 
traced  by  the  radius- vectors  of  the  different  bodies  of  the  system  on  an 
immovable  plane  passing  through  the  fixed  point,  multiplied  respectively 
by  the  masses  of  these  bodies,  will  be  proportional  to  the  time.  This 
theorem  constitutes  the  principle  of  the  CONSERVATION  OF  AREAS. 

"  If  all  the  bodies  are  subject  to  no  other  force  than  their  mutual  action, 
and  there  were  no  center  of  action,  we  could  then  choose  arbitrarily  the 
origin  of  the  radius-vectors,  and  the  preceding  principles  would  be  true 
for  all  points  of  space. 

11  2.  If  a  system  of  bodies  are  subject  to  no  other  force  than  the  mutu- 
al action  of  all  the  parts,  and  are  solicited  by  no  extraneous  force,  the 
common  center  of  gravity  moves  in  a  right  line,  and  uniformly ;  that  is 
to  say,  its  motion  will  be  the  same  as  if  it  obeyed  no  other  force  than  the 
primitive  impulse  communicated  to  it ;  so  that,  in  the  same  manner  as  by 
the  law  of  inertia,  where  a  material  point  cannot,  without  the  intervention 
of  some  extraneous  force,  change  the  motion  it  has  received,  likewise  a 
system  of  bodies  cannot  alter  the  motion  of  its  center  of  gravity  by  the 
single  action  of  its  members  upon  one  another.  This  result  constitutes 
the  principle  of  the  CONSERVATION  OF  THE  UNIFORM  MOTION  OF  THE  CENTER 

OF  GRAVITY. 

"  In  general,  whatever  may  be  the  forces  which  act  on  a  system  of  bo- 
dies, the  motion  of  the  center  of  gravity  of  the  system  is  the  same  as 
though  all  the  bodies  were  united  in  it,  and  all  the  forces  which  solicit  the 
bodies  were  applied  to  it  directly. 

"  3.  The  mass  of  a  body  multiplied  by  the  square  of  its  velocity  is 
called  its  active  force.  If  the  system  under  consideration  is  subject  only 
to  the  mutual  action  of  all  its  members,  and  to  attractions  directed  towards 
fixed  centers,  the  sum  of  the  active  forces  of  the  bodies  which  compose 
the  system,  or  the  entire  active  force  of  the  system,  is  constant  in  the  case 
even  where  several  of  the  bodies  are  compelled  to  move  in  the  direction  of 
given  lines,  or  on  given  surfaces.  This  principle  is  that  of  the  CONSER- 
VATION OF  ACTIVE  FORCES. 

"  4.  The  sum  of  the  active  forces  of  any  system  of  bodies,  during  the 
time  it  employs  in  passing  from  one  position  to  another,  is  a  minimum. 
This  law,  which  was  for  a  long  time  attempted  to  be  derived  from  meta- 
physical considerations,  by  supposing  that  nature  always  employs  in  the 
effects  which  she  produces  the  least  possible  efforts,  results  directly  from 
analytic  formulce,  and  is  called  the  PRINCIPLE  OF  LEAST  ACTION. 

"  The  principle  of  the  conservation  of  areas  holds  good  whatever  may 
be  the  direction  of  the  plane  which  is  chosen  for  the  plane  of  projection  ; 
but  among  all  the  planes  which  can  be  drawn  through  a  given  point,  there 


286  ADDENDA. 

is  always  one  relatively  to  which  the  sum  of  the  areas  traced  by  the  projec- 
tions of  the  radius-vectors,  multiplied  respectively  by  the  masses  of  the 
bodies,  is  a  maximum.  This  plane  possesses  a  very  remarkable  proper- 
ty, namely,  that  this  sum  is  nothing  in  respect  to  every  plane  which  is 
perpendicular  to  it.  This  plane,  moreover,  always  remains  parallel  to 
itself,  whatever  may  be  the  particular  motions  of  each  of  the  bodies  of 
the  system.  Hence  its  position  can  always  be  found, — a  consideration 
which  renders  the  principle  one  of  great  value  in  the  theory  of  the  sys- 
tem of  the  world. 

"  These  four  principles  hold  good  in  the  displacements  of  every  system  of 
bodies,  but  it  is  especially  in  the  motions  of  the  heavenly  bodies  that  their 
truth  is  manifested  by  an  admirable  agreement  of  the  results  of  calcula- 
tion with  those  of  observation.  The  reason  is,  that  the  disturbing  causes, 
such  as  friction,  resistance  of  the  air,  &c.,  which  affect  the  motions  of 
bodies  observed  on  the  earth,  and  which  render  their  mathematical  calcu- 
lation almost  impossible,  disappear,  or  become  nearly  insensible,  in  re- 
spect to  the  heavenly  bodies  circulating  in  the  immensity  of  space,  where 
we  observe  nothing  but  the  effects  of  the  principal  forces  which  animate 
them." 

An  able  writer*  offers  the  following  striking  illustration  of  the  Princi- 
ple of  Least  Action — 

"  Throughout  inanimate  nature,  all  is  done  with  the  least  possible  ac- 
tion ;  no  development  of  force,  however  minute,  is  thrown  away.  If  I 
wished  to  ascend  or  descend  a  hill,  or  pass  from  one  portion  of  it  to  anoth- 
er, with  the  least  possible  muscular  force,  a  slight  consideration  would 
show  me  that  the  precise  path  to  be  pursued,  would  be  dependent  on  the 
form  and  inclination  of  the  different  parts  of  the  hill ;  upon  the  nature  of 
my  own  muscular  energies ;  and  upon  other  data,  of  which  I  could 
scarcely  by  any  possibility  acquire  a  knowledge,  and  on  which,  when 
known,  my  intellectual  powers  would  be  quite  insufficient  to  enable  me 
to  found  a  conclusion.  Under  these  circumstances,  the  chances  are  in- 
finitely greater  that  I  should  select  the  wrong  than  the  right  path.  Now, 
if  I  am  to  project  a  stone  up  the  hill,  or  obliquely  across  it,  or  suffer  it  to 
roll  down  it,  whatever  obstacles  opposed  its  motion,  whether  they  arose 
from  friction,  resistance,  or  any  other  cause,  constant  or  casual,  still 
would  the  stone,  when  left  to  itself,  ever  pursue  that  path  in  which  there 
was  the  least  possible  expenditure  of  its  efforts;  and  if  its  path  were  fix- 
ed, then  would  its  efforts  be  the  least  possible  in  that  path.  This  extra- 
ordinary principle  is  called  that  of  least  action ;  its  existence  and  univer- 
sal prevalence  admit  of  complete  mathematical  demonstration.  Every 
particle  of  dust  blown  about  in  the  air,  every  particle  of  that  air  itself, 

*  Mosely,  "  Mechanics  applied  to  the  Arts,"  Introduction,  p.  xxx. 


ADDENDA.  287 

has  its  motions  subjected  to  this  principle.  Every  ray  oflight  that  passes 
from  one  medium  into  another,  deflects  from  its  rectilinear  course,  that  it 
may  choose  for  itself  the  path  of  least  possible  action  ;  and  for  a  similar 
reason,  in  passing  through  the  atmosphere,  it  bends  itself  in  a  particular 
curve  down  to  the  eye.  The  mighty  planets,  too,  that  make  their  cir- 
cuits ever  within  those  realms  of  space  which  we  call  our  system  ;  the 
comets,  whose  path  is  beyond  it ;  all  these  are  alike  made  to  move  so  as 
best  to  economize  the  forces  developed  in  their  progress." 


ARTICLE  III. 

GREAT  COMET  OF  1843. 
(See  Engraving,  Plate  I.) 

On  the  28th  of  February,  1843,  the  attention  of  numerous  observers, 
in  various  parts  of  the  world,  was  arrested  by  a  comet,  seen  in  the  broad 
light  of  day,  a  little  eastward  of  the  sun.  On  the  day  previous,  indeed, 
it  was  first  noticed  by  Captain  Ray,  at  Conception  in  South  America — 
time,  11  o'clock,  A.M.  Of  the  observations  made  upon  this  remarka- 
ble body  on  the  28th,  the  most  accurate  are  believed  to  be  those  made  at 
Portland,  Maine,  by  Mr.  F.  G.  Clarke.  The  time  of  observation  was 
3h.  2m.  15s.,  mean  solar  time,  and  the  observed  distance,  which  Mr. 
Clarke  thinks  may  be  depended  on  to  10",  was  4°  6'  15",  from  the  farthest 
limb  of  the  sun  to  the  nearest  limb  of  the  comet.  It  resembled  a  white 
cloud  of  great  density,  being  nearly  equally  brilliant  throughout  its  whole 
length,  which  at  this  time  was  estimated  by  different  observers  at  about 
3°.  During  the  first  week  in  March,  the  appearance  of  the  comet  in 
the  southern  hemisphere  was  splendid  and  magnificent,  enhanced,  in  both 
respects,  by  the  transparency  of  a  tropical  sky,  and  the  higher  angle  of 
elevation  above  that  at  which  it  was  seen  by  northern  observers.  At 
Pernambuco,  S.  A.,  on  the  4th  of  March,  it  presented  a  golden  hue  ;  and 
it  was  described  by  the  commander  of  a  ship  as  so  brilliant  as  to  throw  a 
strong  light  on  the  sea. 

At  New  Haven,  it  was  first  seen  after  sunset  on  the  5th  of  March,  and 
by  the  writer  on  the  6th.  It  then  lay  far  in  the  southwest.  On  account 
of  the  presence  of  the  moon,  it  was  not  seen  most  favorably  until  the 
evening  of  the  17th.  It  then  extended  along  the  constellation  Eridanus 
to  the  ears  of  the  Hare,  towards  Sirius,  about  40°  in  length,  slightly  curv- 
ed like  a  goose-quill,  and  colored  with  a  slight  tinge  of  rose  red,  which 
in  a  few  evenings  disappeared,  and  the  comet  afterwards  appeared  nearly 


288  ADDENDA. 

white.  Our  diagram  presents  a  pretty  accurate  idea  of  its  appearance 
on  the  20th  of  March.  Its  nucleus  was  at  this  time  near  the  star  Zibal, 
in  Eridanus,  (R.  A.  46°  4'  38".4,  Dec.  S.  9°  9'  45".5)  and  it  extended 
nearly  parallel  to  the  equator  along  the  constellation  Eridanus,  through 
the  ears  of  the  Hare,  beneath  the  feet  of  Orion,  terminating  near  Sirius. 
The  several  stars  represented  in  or  near  the  comet,  in  the  annexed  dia- 
gram, may  be  easily  identified  by  the  aid  of  a  celestial  globe,  or  a  map 
of  the  stars. 

All  the  astronomers  of  the  age  have  agreed  in  the  opinion,  that  this  is 
one  of  the  most  remarkable  exhibitions  of  a  comet  ever  witnessed,  al- 
though they  are  not  fully  agreed  respecting  the  elements  of  its  orbit,  or 
its  p  riodic  time.  Professor  Pierce  maintains  that  the  most  probable 
period  is  175  years,  and  consequently  that  the  present  is  its  first  return 
since  1668;  but  Messrs.  Walker  and  Kendall  are  of  opinion  that  its  true 
period  is  2  If  years,  and  consequently  that  this  is  the  eighth  return  since 
1668,*  and  that  it  will  visit  our  sphere  again  in  1865. 

This  comet  passed  its  perihelion  on  the  27th  of  February,  at  which 
time  it  almost  grazed  the  surface  of  the  sun,  approaching  nearer  to  that 
luminary  than  any  comet  hitherto  observed.  Its  motions  at  this  time  were 
astonishingly  swift,  and  its  brilliancy  such  as  to  -induce  the  belie/  that  it 
was  at  a  white  heat  throughout  its  whole  extent. 

According  to  the  determination  of  Messrs.  Lauguier  and  Mauvais,  of 
the  Paris  Observatory,  which  is  believed  to  be  as  accurate  as  any,  the 
following  are  the  elements  of  this  comet — 

Time  of  Perihelion.          Lon.  Per.  Lon.  As.  Node.          Inclination.  Per.  Diet.  Course. 

27d.42'291     278°  45' 39"     2°  10'  0"       35°  31'  30"  0.005488      Ret. 


DOUBLE    STARS    AND    NEBULAE. 

(See  Engraving,  Plate  II.) 

As  an  accurate  representation  of  these  delicate  and  interesting  objects 
requires  the  most  finished  engraving  and  printing,  and  cannot  be  well 
executed  in  the  body  of  the  work,  we  annex  a  separate  cut  giving 
a  correct  view  of  several  of  these  remarkable  objects.  The  double 
stars  were  figured  by  the  late  Ebenezer  Porter  Mason,  and  the  nebu- 
la is  from  the  Glasgow  edition  of  Nichols's  "  Architecture  of  the 
Heavens" — a  work  which  contains  a  great  variety  of  elegant  and  accu- 
rate delineations  of  stars  and  nebulae. 


*  See  American  Almanac,  1844,  p.  94.     American  Journal  of  Science,  xlv.  188- 


ADDENDA.  280 


ARTICLE  IV. 

NUMERICAL  RELATIONS  EXISTING  BETWEEN  THE  MEMBERS  OF 
THE  SOLAR  SYSTEM. 

ILLUSTRATED    BY    A    NUMBER    OF    HIGHLY    USEFUL    AND    INTERESTING 
ASTRONOMICAL    PROBLEMS.* 

If  we  contemplate  the  relations  subsisting  between  a  central  body,  as 
the  sun,  and  a  revolving  body,  as  one  of  the  planets,  it  will  be  readily 
understood,  that  if  the  quantity  of  matter  in  the  central  body  is  in- 
creased, while  the  distance  of  the  revolving  body  remains  the  same, 
the  velocity  of  the  revolving  body  must  be  increased  also,  in  order  to 
generate  a  sufficient  centrifugal  force  to  counterbalance  the  increased 
force  of  attraction  in  the  central  body,  arising  from  the  increase  of  its 
mass ;  and  that,  were  the  force  of  attraction  diminished  by  removing 
the  body  to  a  greater  distance  from  the  center,  then  the  rate  of  its  mo- 
tion would  also  have  to  be  diminished,  otherwise  the  centrifugal  force 
would  overpower  the  force  of  attraction.  It  is  a  remarkable  fact,  that 
the  members  of  the  solar  system  are  so  adjusted  to  each  other,  in  re- 
spect to  their  velocities,  distances  from  the  sun,  periodic  times,  and 
gravitation  towards  the  central  body,  that  if  any  one  of  these  particulars 
is  known,  all  the  rest  become  known  also.  Thus,  if  it  were  found  that 
a  new-discovered  planet  moved  in  its  orbit  six  times  as  slow  as  the 
earth,  we  should  know  at  once  that  its  distance  from  the  sun  was  thirty- 
six  times  as  great  as  the  earth's  distance,  that  its  time  of  revolution  was 
two  hundred  and  sixteen  years,  and  that  its  gravitation  towards  the  sun 
was  twelve  hundred  and  ninety-six  times  less  than  that  of  the  earth  ; 
for  the  distance  is  the  square  of  the  number  expressing  the  rate  of  mo- 
tion compared  with  that  of  the  body  taken  as  a  standard ;  the  periodic 
time  is  the  cube  ;  and  the  gravitation  to  the  sun  is  the  biquadrate  of  the 
same  number.  All  this  follows  from  Kepler's  third  law — that  the 
squares  of  the  periodic  times  are  as  the  cubes  of  the  distances;  and 
from  the  law  of  universal  gravitation — that  the  force  of  attraction  is  in- 
versely as  the  square  of  the  distance.  The  four  particulars  named, 
therefore,  constitute  a  series  of  numbers  in  geometrical  progression,  of 
which  the  first  term  is  equal  to  the  ratio.  The  truth  of  this  proposition 
may  be  demonstrated  as  follows. 

*<•  In  the  preparation  of  this  article,  the  author  has  derived  much  assistance  from  a 
small  work,  now  nearly  out  of  print,  containing  the  substance  of  three  lectures  deliv- 
ered to  the  students  of  Yale  College  in  1781,  by  Rev.  Nehemiah  Strong,  at  that  time 
Professor  of  Mathematics  and  Natural  Philosophy. 

37 


290  ADDENDA. 

Let  D  be  the  mean  distance  of  a  planet  from  the  sun,  #  the  ratio  of 
the  diameter  to  the  circumference  of  a  circle,  and  P  the  time  of  revo- 
lution around  the  sun,  or  periodic  time ;  then  the  expression  for  the 

2*D      D  D8 

velocity  is  V=——  x  - .  And  V*  x  p-     But,  by  Kepler's  law,  P2x  D' 

D2  1 

.  •  .  V2  oc  yp  or  Va  x  =r.    Since  a  body  more  remote  from  the  sun  moves 

more  slowly  in  its  orbit  than  a  nearer  body,  and  the  comparative  slow- 
ness, or  retardation,  is  inversely  as  the  velocity,  in  order  to  avoid  frac- 
tional terms,  we  may  put  the  retardation  (R)  in  the  place  of  V,  and  then 
R2xD,  (1.)  If,  therefore,  R  indicates  how  much  slower  a  planet 
moves  than  another,  as  the  earth,  taken  as  a  standard,  the  square  of  R 
will  show  how  much  farther  from  the  sun  the  planet  is  than  the  earth. 

D  D3 

Again,  since  V  oc  — ,  V3  oc  ^.    But,  by  Kepler's  law,  D3oc  P2  .-.  V3x 

^orV'oDp,  andR3ocP(2.) 

Consequently,  if  R  expresses  the  retardation  of  a  planet  in  compari- 
son with  the  earth,  the  cube  of  R  will  express  the  corresponding  peri- 
odic time. 

Finally,  by  the  law  of  gravitation,  the  force  of  gravitation  towards 
the  central  body  varies  as  the  square  of  the  distance  inversely,  or 

G  oc  =-jj.  But  the  diminution  of  gravity  (L)  being  inversely  as  the  grav- 
ity,.L  x  D2 ;  but  D  x  R9  .-.  D2  x  R4,  and  L  x  R4  (3.) 

Therefore,  if  R  denotes  how  much  slower  a  planet  moves  in  its  orbit 
than  the  earth,  R4  will  denote  how  much  less  the  same  body  gravitates 
towards  the  central  body.  Collecting  these  several  results,  it  appears 
that  the  square  of  the  rate  of  motion  gives  the  distance,  its  cube  the  peri- 
odic time,  and  its  fourth  power  the  diminution  of  gravity,  which  numbers 
compose  a  series  in  geometrical  progression  of  which  the  first  term  is  the 
ratio. 

A  number  of  very  useful  and  convenient  rules,  may  be  derived  from 
this  numerical  relation  between  the  members  of  the  solar  system  ;  since, 
when  any  one  of  the  four  things  named  is  given,  all  the  rest  may 
be  found  from  it ;  and  each  of  the  four  may  be  found  in  four  differ- 
ent ways  when  the  other  members  of  the  series  are  given.  This  will  be 
obvious  from  a  few  examples. 

I.  Given  the  RATE  OF  MOTION  or  RETARDATION,  (R.) 

1.  Square  the  retardation  for  the  distance. 

2.  Cube  the  retardation  for  the  periodic  time. 

•3.  Take  the  fourth  power  of  the  retardation  for  the  force  of  gravitation, 


ADDENDA.  291 

II.  Given  the  DISTANCE,  (D.) 

1.  Take  the  square  root  of  the  distance  for  the  rate  of  motion. 

2.  Take  the  cube  of  the  square  root  of  the  distance  for  the  periodic 
time. 

3.  Take  the  square  of  the  distance  for  the  force  of  gravitation. 

III.  Given  the  PERIODIC  TIME,  (P.) 

1.  Take  the  cube  root  of  the  periodic  time  for  the  rate  of  motion. 

2.  Take  the  square  of  the  cube  root  of  the  periodic  time  for  the  dis- 
tance. 

3.  Take  the  biquadrate  of  the  cube  root  for  the  force  of  gravitation. 

IV.  Given  the  diminished  FORCE  OF  GRAVITATION.  (L.) 

1.  Take  the  fourth  root  for  the  rate  of  motion. 

2.  Take  the  square  root  for  the  distance. 

3.  Take  the  cube  of  the  fourth  root  for  the  periodic  time. 

V.  Required  the  RATE  OF  MOTION. 

This  may  be  obtained  by  taking  the  square  root  of  the  distance,  or 
the  cube  root  of  the  periodic  time,  or  the  biquadrate  root  of  the  force  of 
gravitation,  or  by  dividing  the  force  of  gravitation  by  the  periodic  time. 

VI.  Required  the  DISTANCE. 

Take  the  square  of  the  retardation,  or  the  square  of  the  cube  root  of 
the  time,  or  the  square  root  of  the  force  of  gravitation,  or  divide  the 
time  by  the  retardation. 

VII.  Required  the  PERIODIC  TIME. 

We  may  take  the  cube  of  the  retardation,  or  the  cube  of  the  square 
root  of  the  distance,  or  the  cube  of  the  fourth  root  of  the  gravitation,  or 
may  divide  the  gravitation  by  the  retardation. 

VIII.  Required  the  diminished  GRAVITATION. 

It  may  be  found  from  the  fourth  power  of  the  retardation,  or  the 
square  of  the  distance,  or  the  biquadrate  of  the  cube  root  of  the  time, 
or  by  multiplying  the  periodic  time  by  the  retardation. 

According  to  the  foregoing  rules  tables  may  be  formed,  exhibiting,  in 
a  striking,  light,  the  numerical  relations  of  the  members  of  the  solar 
system.  In  the  following  table  the  distances  are  taken  from  Herschel's 
Astronomy,  and  from  these  the  other  particulars  are  determined  by  the 
preceding  rules.  If  Mercury  were  taken  as  the  standard  of  comparison, 
then  the  retardations  of  all  the  other  planets  would  be  greater  than 
unity ;  but,  as  it  is  convenient  to  take  the  earth  as  the  standard,  the 
retardations  of  Mercury  and  Venus  will  be  less  than  unity :  showing 
that  the  velocity  (which  is  expressed  by  the  fraction  inverted)  is  greater 
than  that  of  the  earth.  In  like  manner,  the  force  of  gravitation  of  an 
inferior  planet,  being  greater  than  that  of  the  qprth,  is  the  reciprocal 
of  the  tabular  number. 


292 


ADDENDA. 


TABLE  SHOWING  THE  NUMERICAL  RELATIONS  OF  THE  PRIMARY  PLANETS 


Planets. 

Retardations. 

Distances. 

Per.  Times. 

Force  of  Gravitation. 

Mercury, 

0.62217 

0.38710 

0.24084 

0.14985 

Venus, 

0.85049 

0.72333 

0.61519 

0.52321 

Earth, 

1.00000 

1.00000 

1.00000 

1.00000 

Mars, 

1.23440 

1.52369 

1.88080 

2.32170 

Jupiter, 

2.28100 

5.20277 

11.86700 

27.06900 

Saturn, 

3.08850 

9.53878 

29.46100 

90.98900 

Uranus, 

4.37970 

19.18239 

84.01200 

367.95000 

Neptune, 

5.49040 

30.14512 

165.51000 

908.72000 

PROBLEMS. 

PROB.  1. — The  planet  Pallas  was  discovered  to  have  a  period  of  about 
4j  years.  How  much  slower  does  it  move  in  its  orbit  than  the  earth — 
how  much  further  is  it  from  the  sun — and  how  much  less  does  it  grav- 
itate towards  the  sun  ?  Ans.  R=1.67,  D=2.79,  L— 7.80. 

By  applying  to  the  earth's  motion  per  second,  its  distance  from  the 
sun  in  miles,  and  the  space  through  which  the  earth  departs  in  a 
second  from  a  tangent  to  her  orbit,  the  proportional  numbers  deter- 
mined by  this  problem,  we  may  obtain  the  numerical  value  of  each  of 
these  elements. 

PROB.  2. — -What  would  be  the  periodical  time  of  a  meteor  or  planet 
revolving  close  to  the  earth  ? 

As  the  moon  is  a  body  revolving  around  the  earth  at  a  known  dis- 
tance, and  with  a  known  periodic  time,  it  will  evidently  furnish  the 
necessary  standard  of  comparison.  The  distance  of  the  moon  from  the 
center  of  the  earth  being  60  times  the  earth's  radius,  and  of  course  60 
times  that  of  the  meteor,  its  rate  of  motion  is  ^/  60  times  less.  The 

_3 

retardation  being  -y/60,  the  periodic  time  will  be  602.  Now  what 
part  of  the  moon's  period  is  60^  ?  Divide  the  moon's  period  (27.32 

days)  by  60^,  and  we  have  for  the  answer — 1  hour,  24  minutes,  38.88 
seconds. 

PROB.  3. — What  would  be  the  periodic  time  of  a  body  revolving 
about  the  earth  at  the  distance  of  5000  miles  from  the  center  ?  Ans. 
Ih.  59m.  23.28s. 

PROB.  4. — How  much  faster  must  the  earth  revolve  in  order  that 
bodies  on  its  surface  may  lose  all  their  gravity  ? 

According  to  probler$  1,  the  period  of  a  body  revolving  at  the  sur- 
face of  the  earth,  is  1.4108  hours  ;  and  since,  in  a  circular  orbit,  the 


ADDENDA.  293 

force  of  gravity  and  the  centrifugal  force  are  equal,  therefore,  a  body 
like  that  contemplated  in  problem  1,  is  in  equilibrium  between  these  two 
forces ;  consequently  such  a  body  may  be  considered  as  having  lost  all 
its  gravity,  and  being,  by  the  supposition,  close  to  the  earth,  we  have 
only  to  inquire  how  much  its  velocity  exceeds  that  of  the  earth.  Now 
24  divided  by  1.4108  gives  17.01 ;  which  shows  that  were  the  earth  to 
revolve  on  its  axis  about  17  times  faster  than  it  does  at  present,  the 
bodies  on  the  surface  would  lose  all  their  weight ;  and  were  the  velocity 
greater  than  this,  the  centrifugal  force  would  prevail  over  the  centripe- 
tal, and  the  same  would  fly  off  from  the  earth  in  tangents. 

PROS.  5. — Were  the  moon  to  be  removed  so  far  from  the  earth  as  to 
revolve  about  it  but  once  a  year,  how  much  greater  would  be  its  dis- 
tance than  at  present,  how  much  less  its  velocity,  and  its  gravitation 
towards  the  earth  ? 

Its  period  being  increased  13.37  times,  its  retardation  is  13.37^—2.37  ; 
its  distance  2.372=5.633 ;  and  its  diminished  gravity  5.6332=31.73. 
Or  R=2.37,  D=5.633,  and  L=31.73. 

Multiplying  the  present  distance  of  the  moon,  238,545  miles,  by  5.633, 
we  obtain  about  1,344,000  miles  for  the  distance  at  which  the  moon  must 
have  been  placed  in  order  to  complete  its  revolution  in  one  year. 

PROS.  6. — Were  the  earth's  mass  equal  to  the  sun's,  and  of  course 
354,000  times  as  great  as  at  present,  in  what  time  would  the  moon  re- 
volve  around  it  ? 

Since,  at  a  given  distance,  any  increase  of  mass  in  the  central  body 
is  accompanied  by  an  equal  increase  of  attraction,  which  must  be 
balanced  by  a  corresponding  increase  of  velocity  in  the  revolving  body, 
so  as  to  generate  an  equal  centrifugal  force  ;  and  since  the  centrifugal 
force  in  a  given  circular  orbit  varies  as  the  square  of  the  velocity,  there- 
fore Vs  oc  354,000,  and  V  oc  y'  354,000,  and  the  periodic  time  being  in- 

versely  as  the  velocity,  P  «  -y~00  =594^=  thatis> the  P6™*" 
time  is  diminished  about  595  times  ;  and  27.32  divided  by  595  gives  Ih. 
5m.  57s.  as  the  time  in  which  the  moon  would  be  hurled  around  the 
earth,  were  the  quantity  of  matter  in  the  earth  equal  to  that  in  the 
sun. 

Comets,  in  passing  their  perihelion,  especially  when  that  happens  to 
be  very  near  the  sun,  as  in  the  great  comet  of  1843,  move  with  an  as- 
tonishing rapidity  ;  requiring  a  velocity  not  merely  sufficient  to  gener 
ate  the  centrifugal  force  necessary  to  balance  the  powerful  force  of  at- 
traction  exerted  by  the  sun,  but  greatly  to  exceed  that  force,  since  they 
are  carried  far  without  a  circular  orbit  into  an  elliptical  or  even  a  hy- 
periodic  orbit. 


294  ADDENDA. 

PROB.  7. — How  much  must  the  mass  of  the  earth  be  increased  in 
order  that  the  moon  may  revolve  about  it  in  the  same  time  as  at  present, 
when  removed  to  three  times  her  present  distance  ? 

The  quantity  of  matter  in  the  central  body  is  proportioned  to  the 
cube  of  the  distance  of  the  satellite  divided  by  the  square  of  the  periodic 
time,  (Art.  379.)  Here  the  time  being  given  M  <r  D3— 33=27,  which 
shows  that,  had  the  moon  been  placed  three  times  as  far  from  the  earth 
as  at  present,  she  would  have  required  27  times  as  much  matter  to 
make  the  moon  revolve  in  her  present  period. 

PROB.  8. — How  much  must  the  mass  of  the  earth  be  increased  to 
make  the  moon,  at  her  present  distance,  revolve  in  24  hours  ?  Ans. 
746.4  times. 

PROB.  9. — The  semi-diameter  of  Jupiter  being  11  times  that  of  the 
earth,  and  the  distance  of  its  fourth  satellite  from  the  center  of  the 
planet  being  27  times  the  radius  of  the  planet ;  also  the  sidereal  revolu- 
tion of  the  satellite  being  16.69  days,  while  that  of  the  moon  is  27.3217 
days,  and  her  distance  60  times  the  radius  of  the  earth :  How  much 
does  the  quantity  of  matter  in  Jupiter  exceed  that  of  the  earth  ?  Ans. 
324.49  times. 

PROB.  10. — Suppose  volcanic  matter  to  be  thrown  from  the  moon  to- 
wards the  earth,  required  the  point  where  it  would  be  in  equilibrium 
between  the  two,  the  mass  of  the  moon  being  one-eightieth  that  of  the 
earth  ?  Ans.  24,000  miles  from  the  center  of  the  earth,  nearly. 

PROB.  11. — Suppose  that  the  only  two  bodies  in  the  universe  were  a 
sphere  two  inches  in  diameter,  of  the  same  density  with  the  earth,  for 
the  primary,  and  a  material  point  for  the  satellite :  What  would  be  the 
periodic  time  of  the  satellite,  at  the  distance  of  one  foot,  in  a  circular 
orbit  ?  Ans.  2  days,  9  hours,  41  minutes. 


ARTICLE  V. 
RECENT  DISCOVERIES. 

Within  a  short  period  astronomical  science  has  been  enriched  with 
several  capital  discoveries,  showing  that  neither  the  field  of  observation, 
nor  the  resources  of  mathematical  analysis  are  exhausted,  and  inspiring 
the  hope  and  belief  that  truths  in  astronomy  more  recondite,  and  no  less 
wonderful  than  those  revealed  successively  to  preceding  ages,  will  con. 
tinue  to  crown  the  labors  of  astronomers  to  the  end  of  time. 


ADDENDA.  295 

DISTANCES  OF  THE  STARS. — After  many  fruitless  and  delusory  efforts 
to  measure  the  immense  interval  that  separates  us  from  the  fixed  stars, 
the  great  Prussian  astronomer,  Bessel,  in  the  year  1838,  determined 
this  interesting  and  important  element,  by  observations  on  a  double  star 
in  the  Swan,  (61  Cygni.)  This  star  was  selected  for  the  following 
reasons  :  first,  it  was  known  to  have,  among  all  the  stars,  the  greatest 
proper  motion,  indicating  a  comparatively  great  proximity  to  our  system  ; 
secondly,  being  a  double  star,  it  was  peculiarly  well  adapted  to  instru- 
mental observation ;  thirdly,  situated  as  it  is  among  the  circumpolar 
stars,  observations  could  be  made  upon  it  nearly  every  night  in  the 
year;  and,  finally,  the  great  number  of  small  stars  in  the  imme- 
diate neighborhood,  furnished  the  opportunity  of  selecting  favorable 
stationary  points  from  which  (inasmuch  as  these  more  remote  objects 
might  be  considered  as  entirely  devoid  of  parallax)  any  changes  of 
place  in  the  nearer,  in  consequence  of  an  annual  parallax,  might  be 
readily  estimated.  By  observations  of  the  last  degree  of  refinement, 
conducted  for  a  period  of  several  years,  a  parallax  was  decisively  indi- 
cated, amounting  to  about  one-third  of  a  second  ;  or,  more  exactly,  to 
0/'3483,  implying  a  distance  of  592,200  times  the  mean  distance  of  the 
earth  from  the  sun,  or  a  space  which  it  would  take  light,  moving  at  the 
rate  of  twelve  millions  of  miles  per  minute,  nine  and  a  quarter  years  to 
traverse.*  To  form  some  familiar  notions  of  this  distance,  let  us  sup- 
pose a  railway-car  to  travel  night  and  day,  at  the  rate  of  twenty  miles 
an  hour,  we  should  find  it  would  take  it  about  547  years  to  reach  the 
sun  ;  but  to  reach  61  Cygni  would  require  324,000,000  of  years. 

The  observations  of  Bessel  enabled  him  to  estimate  also  the  period  of 
revolution  of  the  two  stars  composing  the  binary  system  of  61  Cygni, 
and  the  dimensions  of  the  orbit,  and  he  found  the  periodic  time  about 
540  years,  and  the  length  of  the  orbit  about  two  and  a  half  times  that 
of  Uranus.  Knowing  also  the  distance  of  this  star,  we  can  now  deter- 
mine from  its  proper  motion  (five  seconds  a  year)  the  velocity  of  its  mo- 
tion :  this  is  found  to  be  about  forty-four  miles  per  second, — more  than 
double  that  of  the  earth  in  its  orbit — amounting  to  about  one  thousand 
millions  of  miles  per  annum. 

On  account  of  the  smallness  of  the  supposed  parallax  thus  found,  it 
would  not  be  unreasonable  still  to  entertain  a  lingering  suspicion,  that 
it  is  nothing  more  than  the  unavoidable  imperfection  of  instrumental 
measurements,  as  proved  to  be  the  case  in  previous  attempts  to  find 
the  same  element ;  but  the  most  satisfactory  evidence  which  the  world 


•  It  will  be  remarked  that  this  is  the  result  of  the  second  series  of  observations,  and 
M  deemed  more  accurate  than  those  mentioned  in  the  text 


296  ADDENDA. 

can  have  that  such  is  not  the  fact  in  the  present  instance,  but  that  the 
parallax  is  truly  found,  is  that  the  most  celebrated  astronomers  of  the 
age,  after  rigorous  scrutiny,  have  acknowledged  the  reality  and  sound- 
ness  of  the  determination. 

Several  other  stars  have  of  late  been  supposed  to  indicate  a  parallax ; 
and  one  of  them,  Alpha  Centauri,  is  thought  by  some  to  be  nearer  to  us 
than  61  Cygni,  having  a  parallax  of  nine-tenths  of  a  second.  The 
great  Russian  astronomer,  M.  Struve,  has  also  announced  a  parallax  in 
Alpha  Lyra  of  a  quarter  of  a  second ;  but  this  result  has  not  yet  been 
confirmed. 

NEW  PLANETS. — Unexpectedly,  four  new  planets  have  in  rapid  suc- 
cession been  revealed  to  us,  namely — Astrsea,  Iris,  Hebe,  and  Neptune. 
The  first  three  belong  to  the  group  of  Asteroids,  making  the  whole  num- 
ber at  present  known,  seven  instead  of  four.  They  are  very  small 
bodies,  seen  by  the  telescope  as  faint  stars,  but  are  known  to  be  planets 
from  their  having  a  motion  of  revolution  around  the  sun.' 

The  discovery  of  the  planet  Neptune,  by  a  distinguished  French  as- 
tronomer, Le  Verrier,  is  one  of  the  most  remarkable  events  in  astrono- 
my ;  both  its  existence  and  its  position  in  the  heavens  having  been  re- 
vealed to  a  profound  mathematical  analysis  before  it  was  seen  with  the 
telescope.  The  method  of  investigation,  although  laborious  and  intri- 
cate, is  not  difficult  to  be  understood,  but  may  be  described  in  very  sim- 
ple terms.  The  planet  Uranus  has  been  long  known  to  be  subject  to 
certain  irregularities  in  its  revolution  around  the  sun,  not  accounted  for 
by  all  the  known  causes  of  perturbation.  In  some  cases  the  deviation 
from  the  trqe  place,  as  given  by  the  tables,  differs  from  actual  observa- 
tion two  minutes  of  a  degree, — a  quantity,  indeed,  which  seems  small, 
but  which  is  still  far  greater  than  occurs  in  the  case  of  the  other  planets, 
Jupiter  and  Saturn,  for  example,  and  far  too  great  to  satisfy  the  extreme 
accuracy  required  by  modern  astronomy.  This  long  since  suggested 
to  astronomers  the  possibility  of  one  or  more  additional  planets,  hitherto 
undiscovered,  which,  by  their  attractions,  exerted  on  Uranus  a  great 
disturbing  influence. 

Le  Verrier,  assuming  the  existence  of  such  a  planet,  applied  him- 
self, by  the  aid  of  the  calculus,  guided  by  the  law  of  universal  gravi- 
tation, to  the  inquiry,  where  is  it  situated — at  what  distance  from  the 
sun — and  in  what  point  of  the  starry  heavens  ?  According  to  Bode's 
law  of  the  planetary  distances,  (according  to  which  Saturn  is  nearly 
twice  as  far  from  the  sun  as  Jupiter,  and  Uranus  twice  as  far  as  Saturn,) 
he  inferred  that,  if  a  planet  exists  beyond  Uranus  its  distance  is  proba- 
bly twice  that  of  Uranus,  or  about  thirty-six  millions  of  miles  from  the 
sun,  which  is  about  thirty-eight  times  that  of  the  earth.  He  finally 


ADDENDA.  297 

fixed  it  thirty-six  times  the  mean  distance  of  the  earth.  Hence  its 
periodic  time  would  be  216  years.  After  reasoning  from  analogy,  and 
the  doctrine  of  universal  gravitation,  respecting  the  position  and  mass 
which  a  body  must  have  in  order  to  occasion  the  perturbations  of  Uranus 
to  be  accounted  for,  equations  were  formed  between  these  perturbations 
and  the  elements  of  the  body  in  question,  both  known  and  unknown. 
These  equations  involved  nine  unknown  quantities,  and  their  resolution 
involved  difficulties  almost  insurmountable  ;  but,  by  the  most  ingenious 
artifices,  the  several  unknown  quantities  were  successfully  eliminated, 
either  directly  or  by  repeated  approximations,  until  he  arrived  at  ex- 
pressions for  the  elements  of  the  unknown  planet,  which  gave  its  exact 
place  among  the  stars,  its  quantity  of  matter,  the  shape  of  its  orbit,  and 
the  period  of  its  revolution.  At  the  sitting  of  the  French  Academy, 
August  31,  1846,  Le  Verrier  presented  the  following  elements  :  — 

Longitude  of  the  planet,  Jan.  1,  1847,        -  326°  32" 

Mass,  that  of  the  sun  being  1,   - 


Eccentricity,  -  0.107 

Time  of  revolution,  -  -  -  217.387  years 

Longitude  of  the  perihelion,      -  -  284°  45' 

Major  axis  of  the  orbit,  that  of  the  earth  being  1,  36.154 

He  was  therefore  enabled  to  say,  that  the  planet  was  then  just  passing 
its  opposition,  and  consequently  was  most  favorably  situated  for  obser- 
vation, and  on  account  of  the  slowness  of  its  motion,  would  remain  in  a 
very  favorable  position  for  three  months  afterwards.  Le  Verrier  wrote 
to  M.  Galli,  of  Berlin,  communicating  his  latest  results,  and  requesting 
him  to  reconnoiter  for  the  stranger,  directing  his  telescope  to  a  point 
about  five  degrees  eastward  of  the  well-known  star,  Delta  Capricorni. 
That  astronomer  no  sooner  pointed  his  telescope  to  the  region  assigned, 
than  he  at  once  recognised  the  body,  its  place  being  only  52  minutes  of 
a  degree  distant  from  the  position  marked  out  for  it  by  Le  Verrier,  and 
its  apparent  diameter  being  almost  the  same  that  he  had  assigned. 

By  a  singular  coincidence,  a  young  mathematician  of  the  University 
of  Cambridge,  (Eng.,)  Mr.  Adams,  had,  without  the  least  knowledge  of 
what  M.  Le  Verrier  was  doing,  arrived  at  the  same  great  result.  But 
having  failed  to  publish  his  paper  until  the  world  was  made  acquainted 
with  the  facts  through  the  other  medium,  he  has  lost  much  of  the  honor 
which  the  priority  of  discovery  would  have  gained  for  him. 

Thus  two  distinguished  mathematicians,  unknown  to  each  other,  and 
by  entirely  independent  processes,  had  arrived  at  the  same  results,  as 
regarded  both  the  existence  of  the  supposed  planet,  and  the  region  of 

38 


298  ADDENDA. 

the  starry  heavens  where  at  that  moment  it  lay  concealed ;  and,  to 
crown  all,  astronomers,  in  obedience  to  the  directions  of  one  of  them, 
had  pointed  their  telescopes  to  the  spot  and  found  it  there.  The  con- 
viction on  the  mind  of  every  one  was,  that  nothing  but  absolute  truth 
could  abide  a  test  so  unequivocal.  It  still  remained,  however,  to  deter- 
mine  by  observation  whether  the  body  actually  conformed,  in  all  re- 
spects, to  the  results  of  theory.  To  settle  this  point  completely,  that  is, 
to  determine  with  precision  the  elements  of  the  orbit  from  observation, 
would  require  a  long  time  in  a  planetary  body  whose  motion  was  so 
slow  that  more  than  two  centuries,  as  was  supposed,  would  be  required 
to  complete  a  single  revolution.  But  if  it  should  be  found,  that  among 
preceding  catalogues  of  the  stars  this  body  might  have  been  included, 
and  its  place  recorded  as  a  fixed  star,  then,  by  comparing  that  place 
with  its  present  position,  and  noting  the  interval  of  time  between  the 
two  observations,  we  might  thus  learn  the  rate  of  its  motion  and  its 
periodic  time,  and  might  thence  deduce  various  other  particulars  de- 
pendent on  these  elements.  Our  distinguished  countryman,  Mr.  S.  C. 
Walker,  then  connected  with  the  observatory  at  Washington,  undertook 
this  investigation.  First,  from  the  observations  already  accumulated 
he  calculated  the  path  which  the  planet  must  have  pursued  for  the  last 
fifty  or  sixty  years,  and  by  tracirig  this  path  among  the  stars  of  La- 
lande's  catalogue,  he  found  that  it  passed  within  two  minutes  of  a  star 
of  the  seventh  magnitude,  which  was  recorded  as  being  seen  in  May, 
1795.  Professor  Hubbard,  of  the  same  observatory,  on  reconnoitering 
for  this  star,  ascertained  that  it  was  missing.  Little  doubt  remained, 
that  the  star  seen  by  Lalande  was  the  planet  of  Le  Verrier ;  and  this 
conclusion  was  confirmed  by  calculating  its  orbit  on  this  supposition, 
and  comparing  the  results  with  the  places  it  has  actually  occupied  since 
it  fell  within  the  sphere  of  observation. 

The  results  thus  obtained  were,  however,  materially  different  from 
those  of  Le  Verrier  and  Adams.  Instead  of  a  period  of  216  years,  as 
given  by  Le  Verrier,  they  gave  a  period  of  only  166  years ;  and  in- 
stead of  a  distance  of  3600  millions  of  miles,  the  new  period  would  re- 
quire a  distance  of  only  2364  millions.  The  eccentricity  of  the  orbit, 
moreover,  according  to  Walker,  was  much  less  than  had  been  assigned 
to  it,  the  orbit  being,  in  fact,  very  nearly  circular,  while  by  Le  Verrier's 
estimate  it  was  considerably  elliptical.  The  longitude,  in  fact,  proved 
to  be  nearly  the  same  as  that  assigned ;  and  hence  the  fortunate  dis- 
covery of  the  body  by  the  telescope. 

Professor  Pierce,  the  able  professor  of  astronomy  in  Harvard  Univer- 
sity, has  been  led  to  the  conclusion,  both  from  his  own  theoretical  in- 
vestigations and  from  the  investigations  of  Mr.  Walker,  that  this  planet 


ADDENDA.  299 

does  not  fully  account  for  the  residuary  perturbations  of  Uranus ;  and, 
therefore,  that  its  discovery  at  the  place  assigned  by  Le  Verrier,  was 
rather  the  result  of  accident  than  the  legitimate  consequence,  of  his 
analysis. 

CENTER  OF  THE  UNIVERSE. — At  the  end  of  the  preceding  treatise  we 
have  suggested  reasons  for  believing  that  the  whole  host  of  heaven  re- 
volve around  a  common  center.  Dr.  Mcedler,  of  the  Imperial  Observa- 
tory at  Dorpat,  has  recently  not  only  asserted  this  doctrine,  but  has  en- 
deavored to  show  the  exact  position  of  that  center.  He  fixes  it  in  the 
Pleiades,  and  asserts  that  Alcyone,  the  brightest  star  of  this  group,  is  the 
true  "  central  sun,"  around  which  all  the  stars  of  our  visible  firmament 
revolve,  in  obedience  to  the  law  of  universal  gravitation.  The  proofs 
of  this  remarkable  hypothesis  are  deemed  too  incomplete,  at  present,  to 
command  entire  assent ;  but  the  method  of  investigation  pursued  by  this 
distinguished  astronomer,  opens  a  new  field  of  observation  and  of  specu- 
lation, and  promises  to  lend  a  new  interest  to  inquiries  into  the  mechan- 
ism of  the  universe. 

REVELATIONS  OF  THE  TELESCOPE. — Practical  astronomy  has  of  late 
been  enriched  with  a  number  of  great  telescopes,  which  have  disclosed 
new  wonders  in  the  starry  heavens.  The  most  remarkable  of  these 
are  the  grand  reflector  constructed  by  Lord  Rosse,  an  Irish  nobleman, 
and  the  great  refractors  belonging  respectively  to  the  Pulkova,  the  Cin- 
cinnati, and  the  Cambridge  observatories. 

Lord  Rosse 's  telescope  considerably  exceeds  in  dimensions  and  in 
power  the  great  40  feet  reflector  of  Sir  William  Herschel,  being  fifty 
feet  in  focal  length  and  having  a  diameter  of  six  feet,  whereas  that  of 
the  Herschelian  telescope  was  only  four  feet.  This  unexampled  mag- 
nitude makes  this  instrument  superior  to  all  others  in  light,  and  fits  it 
pre-eminently  for  observations  on  the  most  remote  and  obscure  celestial 
objects,  as  the  faintest  nebulae.  But  its  unwieldy  size,  and  its  liability 
to  loss  of  power  by  the  tarnishing  or  temporary  blurring  of  the  great 
speculum,  will  render  it  far  less  available  for  actual  research  than  the 
great  refractors  that  come  in  competition  with  it. 

Until  recently,  it  was  thought  impossible  to  form  perfect  achromatic 
object-glasses  of  more  than  about  five  inches  diameter ;  but  they  have 
been  successively  enlarged,  until  we  can  no  longer  set  bounds  to  the 
dimensions  which  they  may  finally  assume.  The  Pulkova  telescope 
(at  St.  Petersburg)  has  a  clear  aperture  of  about  fifteen  inches,  and  a 
focal  length  of  twenty-two  feet.  That  of  Cincinnati  is  somewhat  small- 
er, its  object-glass  being  twelve  inches  diameter,  and  its  length  seven- 
teen  feet ;  but  in  power  it  is  supposed  to  be  fully  equal  to  the  Russian 
instrument.  The  telescope  recently  acquired  by  Harvard  University, 


300  ADDENDA. 

is  perhaps  the  finest  refractor  hitherto  constructed.  Its  dimensions  are 
nearly  the  same  with  those  of  the  Pulkova  instrument,  but  its  perform- 
ances  are  thought  to  be  superior  even  to  that. 

These  magnificent  telescopes  have  afforded  views  of  celestial  objects, 
more  splendid  and  exciting  than  any  previously  enjoyed  by  man.  In  a  sci- 
entific point  of  view,  the  most  interesting  of  these  revelations  consist  in  the 
resolution  of  nebulae  before  deemed  irresolvable,  and  thus  countenancing 
the  idea  that  this  term  is  applicable  only  to  the  comparative  powers  of 
our  instruments;  that,  if  any  objects  of  this  class  remain  unresolved,  it 
will  only  be  because  the  telescope  has  not  yet  acquired  the  requisite 
power  to  separate  them  into  stars.  Under  these  mighty  instruments, 
what  was  before  a  faint  wisp  of  fog  on  the  confines  of  creation,  expands 
suddenly  into  innumerable  suns,  composing  a  glorious  firmament  of 
stars.  The  Cambridge  telescope  has  succeeded  in  the  resolution  of  the 
great  nebula  of  Orion  more  complete  than  had  been  effected  even  by 
Lord  Rosse's  "  Leviathan"  Reflector,  and  is  thus  proved  to  be  one  of  the 
finest  instruments  (probably  the  finest  refractor)  in  the  world. 


OUTLINES 


OP  A 


COURSE    OF   LECTURES 


OK 


ASTRONOMY, 


ADDRESSED    TO   THE   SENIOR   CLASS 

Df 

TA1E  COLLEGE. 


BY  DENISON  OLMSTED,  LL.D., 

PROFESSOR   OF   NATURAL    PHILOSOPHY   AND   ASTRONOMY. 

1850. 


JK3T  THESE  LECTURES  are  delivered  to  the  Class  after  they  have 
finished  their  recitations  in  the  Text-book,  and  are  therefore  presumed  to 
be  acquainted  with  the  Elements  of  Astronomy.  To  persons  attending 
the  course,  who  have  not  read  this  or  some  similar  work,  it  is  recom- 
mended to  accompany  the  Lectures  with  the  perusal  of  the  author's 
"  Letters  on  Astronomy." 

In  the  delivery  of  the  Lectures,  the  pupils  are  supposed  to  have  these 
Outlines  before  them,  and  to  accompany  the  lecturer  through  the  suc- 
cessive heads  of  each  Lecture.  They  will  afterwards  form,  the  basis  of 
the  examinations,  both  public  and  private,  on  the  same  subjects. 


OUTLINES 


OP 


LECTURES  ON  ASTRONOMY, 


LECTURE   I. 

INTRODUCTORY. 

PLAN  OF  THE  COURSE — To  consist  chiefly  of  an  exposition  of 
the  Structure  of  the  Universe — Prefaced  with  a  concise  view  of 
the  History  of  the  Science — Meaning  of  the  phrase  "  Structure 
of  the  Universe,"  and  how  this  subject  differs  from  the  elements 
of  the  science  contained  in  the  text-book — here  all  things  consid- 
ered in  their  relations  to  one  grand  whole,  the  System  of  the 
World.  More  interesting  than  the  mere  study  of  the  elements, 
as  the  study  of  the  classics  is  more  interesting  than  that  of  gram- 
matical rules,  which  are  only  preparatory.  Things  usually  more 
interesting  in  their  relations,  than  in  their  individualities. 

Subjects  of  the  present  lecture,  the  different  classes  of  astrono- 
mers— Laws  of  reasoning  in  astronomy — Pleasures  and  advan 
tages  of  the  study  of  the  heavenly  bodies. 

I.  DIFFERENT  CLASSES  OF  ASTRONOMERS. — Great  diversity  and 
separate  nature  of  the  gifts  that  have  characterized  different  as- 
tronomers— these  to  be  considered  in  estimating  the  respective 
authority  of  each. 

1.  The  meditative  class,  as  Pythagoras,  Copernicus. 

2.  The  mechanical  classes  Tycho  Brahe,  Sir  William  Herschel. 
2.  The  mathematical  class,  as  La  Place,  Le  Verrier. 

4.  The  class  of  consummate  astronomers — those  who  have 
united  these  different  qualities,  as  Kepler,  Galileo,  Newton. 

II.  LAWS  OF  REASONING  IN  ASTRONOMY. — Danger  of  resting  on 
slight  evidences  or  feeble  analogies,  when  it  is  supposed  full  evi- 
dence is  unattainable — Unreasonableness  of  skepticism  and  incre- 
dulity in  regard  to  the  established  truths  of  the  science.  Advan- 
tages of  having  in  view  a  few  leading  principles  of  reasoning. 

1.  The  possibility  of  a  thing  but  slight  evidence  of  its  reality. 

2.  The  same  causes  produce  the  same  effects — proper  meaning 
and  restrictions  of  this  law. 


4  OUTLINES    OP    LECTURES 

3.  The  argument  from  analogy  explained — its  use  and  abuse 
in  astronomy. 

4.  The  argument  from  authority — to  be  received  with  caution, 
and, of  each  astronomer  in  his  own  peculiar  sphere  of  excellence, 

III.  ADVANTAGES  OF  THE  STUDY. 

1.  To  commerce  and  navigation. — Stars  earlier  guides  than  the 
compass — Now  far  more  accurate — Example  of  the  accuracy  of 
the  lunar  method  given  by  Basil  Hall — Constancy  of  the  motions 
of  the  heavenly  bodies,  and  the  perfection  of  instrumental  meas- 
urement, as  asserted  by  Dr.  Bowditch. 

2.  To  chronology. — Use  of  the  precession  of  the  equinoxes,  of 
solar  and  lunar  eclipses,  of  the  situation  of  the  pole  of  the  earth 
among  the  stars  for  fixing  remote  dates.     Command  which  the 
astronomer  has  over  time,  past  and  future. 

3.  In  furnishing  standards  of  weights  and  measures. — Impor- 
tance of  this  subject  in  business  transactions — Difficulty  of  finding 
perfect  standards — Standards  derived  from  an  arc  of  the  meridian, 
and  from  the  pendulum. 

4.  Intellectual  advantages. — Fulfils  the  two  great  purposes  of 
education — to  enlarge  the  mental  powers,  and  to  store  the  mind 
with  important  truths.     Employs  a  variety  of  faculties — stimulates 
to  new  efforts — induces  habits  of  profound  reflection — favorable 
to  the  development  of  both  the  imagination  and  the  intellect. 

5.  Moral  advantages. — What  class  of  astronomers  have  been 
devout,  and  what  class  undevout  men — tendency  of  the  contem- 
plation of  the  heavens  to  awaken  devotional  feelings — Characters 
exhibited  by  great  astronomers — tendency  to  inspire  devotion 
and  the  love  of  truth  noticed  by  the  ancients,  as  Lucretius  and 
Cicero — modesty  of  great  astronomers  asserted  by  Chalmers. 

IV.  PLEASURES  OF  THE  STUDY. 

1.  Delight  experienced  in   contemplating  the  starry  heavens 
with  all  the  lights  of  modern  astronomy. 

2.  Pleasure  of  recognising  known  constellations  when  in  for- 
eign lands. 

3.  Interesting  reflections  connected  with  the  immutability  of 
the  heavens. 

4.  Fascinating  nature  of  the  study,  both  to  the  mathematician, 
who  discerns  new  laws,  and  to  the  practical  astronomer,  who 
discovers  new  worlds. 

LECTURE  II. 

ANCIENT    ASTRONOMY. 

Two  elaborate  histories  of  astronomy  by  Bailly  and  Delambre. 
Four  ancient  nations  that  cultivated  this  science — antiquity  of  the 
study — two  leading  objects,  Eclipses  and  Astrology. 


ON    ASTRONOMY. 


s 


Eclipses. — Importance  attached  to  them — supposed  connection 
with  impending  disasters — how  regarded  among  the  Chinese. 

Astrology. — Objects — rank  of  astrologers — pretensions  of  the 
art — horoscope. 

Chinese  and  Indian  Nations. — High  pretensions  to  antiquity 
— Eclipses  recorded  from  a  very  high  antiquity — Obliquity  of  the 
ecliptic  determined — Astronomy  of  the  Brahmins. 

Chaldeans. — Excellence  of  their  observations — Discovery  of 
the  Saros. 

Egyptians. — Cardinal  points  indicated  by  the  sides  of  the  Pyr- 
amids— astronomical  paintings  and  inscriptions. 

Greeks. — Long  after  the  preceding  nations — three  successive 
schools,  at  Miletus,  Crotona,  and  Alexandria.  How  the  sages  of 
Greece  acquired  their  knowledge — how  the  modes  of  instruction 
in  those  ancient  schools  differed  from  modern  methods — Tholes — 
doctrines  of  his  school. 

Pythagoras. — Extent  and  celebrity  of  his  school — its  date — 
Great  truths  foreshadowed  to  Pythagoras — Singular  opinions  en- 
tertained by  him  respecting  the  music  of  the  spheres.  State  of 
astronomy  in  his  time. 

Alexandrian  School. — Date — by  whom  established — great  as- 
tronomers connected  with  it — Hipparchus — greatest  astronomer 
of  antiquity — when  he  flourished,  and  where — his  discoveries — in- 
struments— trigonometry,  plane  and  spherical — tables  of  the  sun 
and  moon — eccentricity  of  the  solar  and  lunar  orbits — precession 
of  the  equinoxes — motion  of  the  apsides — backward  motion  of  the 
moon's  nodes — exact  length  of  the  year,  and  obliquity  of  the 
ecliptic  —  his  mode  of  calculating  eclipses  —  catalogue  of  the 
stars. 

Crystalline  spheres  of  Eudoxus. 

Ptolemy. — Greatest  writer  among  the  ancient  astronomers — 
Almagest — Ptolemaic  system  of  astronomy. 

Romans. — The  little  attention  they  paid  to  astronomy. 

Arabians. — The  astronomers  of  the  Middle  Ages. 


LECTURE  III. 

MODERN    ASTRONOMY. FROM    COPERNICUS    TO    NEWTON. 

COPERNICUS,  1473. — State  of  the  age — Class  to  which  he  be- 
longed— His  system  of  the  world — How  his  merits  exceed  those 
of  Pythagoras — State  of  the  science  at  that  age — His  proofs  de- 
fective— His  system  compared  with  that  of  Eudoxus,  and  with 
that  of  Ptolemy. 

TYCHO  BRAKE,  1546-1473. — Seventy-three  years  after  Coper- 
nicus— Birth  and  education — Class  to  which  he  belonged — His 


6  OUTLINES    OF    LECTURES 

observatory— Number  and  value  of  his  observations — What  in- 
struments he  had,  and  what  he  was  destitute  of— His  removal  to 
Prague — his  weaknesses. 

KEPLER,  1571-1546. — Twenty-five  years  younger  than  Tycho 
Brahe — Birth  and  education — Mysterium  Cosmographicum,  its 
object  and  character — Characteristics  of  his  genius — Alliance 
with  Tycho.  His  Laws — -the  first  discovered  in  astronomy — their 
importance.  His  eccentricities. 

GALILEO,  1564-1546. — Eighteen  years  younger  than  Tycho, 
seven  years  older  than  Kepler — Early  mechanical  studies — In- 
vention of  the  telescope — discoveries  with  it — confirmation  of  the 
Copernican  system — Disputes  with  the  Aristotelians — persecu- 
tions— Character  of  his  genius — a  "  consummate"  philosopher. 

BACON. — Cotemporary  with  Galileo — their  respective  merits 
compared  in  philosophy  and  astronomy. 


LECTURE  IV. 

MODERN    ASTRONOMY. FROM    SIR    ISAAC    NEWTON    TO    THE 

PRESENT    TIME. 

NEWTON,  1642-1564. — Seventy-eight  years  later  than  Gal- 
ileo— his  rank  among  the  human  race — Saying  of  Bailly — 
Characteristics  of  his  genius — Three  capital  discoveries — Eight 
great  discoveries  of  the  17th  century,  1.  Pendulum.  2.  Tele- 
scope. 3.  Logarithms.  4.  Micrometers.  5.  Algebra  applied 
to  Geometry.  6.  Kepler's  Laws.  7.  Calculus.  8.  Universal 
Gravitation.  Newton's  inquiries  into  the  philosophy  of  Light 
and  Colors.  Discovery  of  the  Law  of  Gravitation — Why  this  is 
esteemed  the  most  important  principle  in  physics — What  it  has 
done  for  the  cause  of  truth  [accounts  for  the  celestial  motions — 
weighs  the  sun  and  planets — determines  the  exact  figure  of  the 
earth,  and  of  every  body  in  the  solar  system — accounts  for  all  the 
irregularities  in  the  motions  of  the  heavenly  bodies — accounts  for 
the  tides,  and  determines  their  height  at  every  place — suggests 
new  fields  of  observation — anticipates  the  discoveries  of  the  tele- 
scope, and  corrects  the  most  refined  observations — determines 
the  stability  of  the  solar  system — predicts  the  exact  return  of 
comets — and  reveals  to  us  new  planets.]  Whether  the  doctrine 
of  universal  gravitation  will  ever  be  superseded. 

Personal  qualities  and  virtues  of  Newton. 

OBSERVATORIES. — Greenwich — Astronomers  royal — Extent  and 
accuracy  of  their  observations — importance  of  these. 


ON    ASTRONOMY. 


SOCIETIES. — Royal  Society  of  London — Academy  of  Sciences 
at  Paris — Astronomical  Society  of  London. 

Astronomers  of  England  and  France — Respective  excellence 
of  each — Other  great  astronomers  of  Europe — State  of  astronomy 
in  our  own  country. 


LECTURE  V. 

THE    SUN. 

Remaining  part  of  the  course  intended  to  explain  the  structure 
of  the  universe — Embraces  two  inquiries,  (1)  What  bodies  com- 
pose the  universe,  (2)  How  they  are  arranged  so  as  to  form  the 
system  of  the  world.  Order  of  subjects,  the  Sun,  Moon,  Planets, 
Comets,  Stars,  and  Nebulae — Mechanism  of  the  Universe. 

SUN. — Important  relations  to  us  as  the  source  of  light,  heat,  and 
attraction — Consequences  were  the  sun  withdrawn. 

Diurnal  Revolution. — Phenomena  at  the  equator,  and  in  differ- 
ent latitudes— Oppressiveness  of  his  direct  rays — seems  directly 
overhead  throughout  the  torrid  zone — Opinion  of  the  ancients  that 
the  torrid  zone  was  uninhabitable — Provisions  of  nature  for  miti- 
gating the  intensity  of  heat.  Phenomena  of  different  latitudes 
from  the  equator  to  the  pole — at  mid-summer — at  the  equinoxes 
— at  mid-winter — Appearances  of  the  sun  at  these  different  sta- 
tions at  different  seasons  of  the  year — Phenomena  of  twilight  in 
various  countries — lasts  all  night  at  midsummer  beyond  48 1°  of 
latitude — length  of  twilight  in  our  latitude — length  of  the  shortest 
night — ditto  of  the  longest  night.  In  travelling  round  the  earth 
eastward,  a  day  gained — westward,  a  day  lost — Case  of  two  trav- 
ellers going  round  the  earth,  and  returning  to  the  same  place — 
also,  when  meeting  at  some  other  point,  as  at  the  Sandwich  Islands 
— Case  of  ships,  and  of  missionaries — Effect  of  these  principles  on 
the  Sabbath — When  the  Sabbath  begins — Case  where  the  twilight 
lasts  all  night — where  it  is  continual  day — where  the  parties  meet 
on  opposite  sides  of  the  earth.  When  is  the  design  of  the  ordi- 
nance fulfilled  ? — Reasons  for  consulting  expediency. 


LECTURE  VI. 

THE    SUN. 

ANNUAL  REVOLUTION. — Mode  of  distinguishing  between  the 
diurnal  and  annual  motions  of  the  sun — cause  of  each  explained 
— Compatibility  of  the  two  motions — Solar  days  longer  in  winter 
than  in  summer — Sun  eight  days  longer  on  the  north  than  south 


8  OUTLINES    OP    LECTURES 

of  the  equator — why — Greater  nearness  in  winter  compensated  by 
the  shorter  time,  so  as  to  preserve  an  equality  in  the  distribution 
of  heat. 

Obliquity  of  the  Ecliptic. — Modes  of  finding  it  practised  in 
ancient  and  in  modern  times— Its  amount  recorded  at  different 
epochs — Its  annual  variations — Present  amount. 

ROTATION  OF  THE  SUN. — Period — Telescopic  appearance  of 
the  sun. 

SPOTS. — Description — extent — cause. 

Galileo's  hypothesis. — Fiery  sea — Volcanic  scoriae — objections. 

Lalande's. — Recession  of  the  fiery  sea — Solar  mountains  and 
valleys — objections. 

HerscheFs  views  of  the  nature  and  constitution  of  the  sun — 
Supposed  atmospheres — whence  the  light  and  heat — false  reason- 
ing— whether  the  surface  of  the  sun  is  in  a  state  of  combustion  or 
of  ignition. 


LECTURE  VII. 

• 

THE    MOON. 

Its  uses  for  light,  for  tides,  for  months,  and  for  longitudes — • 
comparative  nearness — Distances  of  the  other  heavenly  bodies 
measured  by  millions,  of  the  moon  by  thousands — Appearances 
to  the  naked  eye  of  its  phases,  and  of  its  light  and  shade,  when  full. 

REVOLUTION  AROUND  THE  EARTH. — Changes  in  its  path,  its  ve- 
locity, its  nodes — Numerous  irregularities — their  general  cause — 
why  so  much  more  numerous  in  the  moon  than  in  any  other  of 
the  heavenly  bodies — Pains  taken  to  perfect  the  lunar  tables — 
different  means  employed  for  this  purpose. 

TELESCOPIC  APPEARANCES  OF  THE  MOON. — Directions  for  view- 
ing the  moon — Magnifying  powers  employed  for  different  pur- 
poses— Successive  appearances  at  different  ages — Best  view  at 
quadrature.  Objects  to  be  particularly  noted. 

1.  Broken  surface — extreme  irregularity. 

2.  Appearance  of  the  terminator — Single  mountains  described 
— Height  of  the  lunar  mountains. 

3.  Circular  chains — long  ridges. 

4.  Radiations  from  Tycho,  Kepler,  and  Copernicus. 

5.  Valleys  and  craters — resemblance  of  these  to  volcanic  cra- 
ters on  the  earth — mode  of  formation. 

WHETHER  THE  MOON  HAS  AN  ATMOSPHERE? — Want  of  .evidence 


ON    ASTRONOMY. 

— None  seen  in  eclipses — Supposed  appearance  of  twilight — No 
change  of  place  by  refraction  in  a  star  undergoing  occupation — 
Atmosphere,  if  any,  very  small  and  rare. 

WHETHER  THERE  IS*WATER  IN  THE  MOON  ? — Places  named  seas 
not  such — no  clouds  nor  fogs — no  atmosphere  of  watery  vapor. 

VOLCANOES. — Proofs  of  their  existence — objections  considered. 

LUNAR  INFLUENCES. — Supposed  influence  on  the  weather — on 
diseases — on  vegetables. 

Whether  we  can  ever  hope  to  see  lunar  inhabitants  or  their 
works  ? — Limits  to  the  powers  of  the  telescope.  Effect  of  apply- 
ing different  magnifiers  from  240  to  10,000. 


LECTURE  VIII. 

THE    PLANETS. 

In  nearness,  next  to  the  sun  and  moon — the  sun,  moon,  and 
planets,  compose  one  family  in  the  stellar  universe.  Influence  of 
the  planets  upon  our  world  slight,  though  considered  in  the  days 
of  astrology  as  very  great. 

NUMBER. — How  many  were  known  to  the  ancients — Analogy 
traced  by  Kepler  to  the  five  regular  solids — Whole  number,  in- 
cluding the  earth  and  moon,  seven — Supposed  by  the  Aristotelians 
to  be  necessarily  seven — Reasoning  of  the  Roman  doctor  against 
Galileo's  discovery  of  Jupiter's  satellites.  Uranus  added  in  1781 
— Ceres,  Pallas,  Juno,  and  Vesta,  about  the  commencement  of  the 
present  century — more  recently,  Astra3a,  Hebe,  Iris,  and  Neptune. 

Remarkable  discovery  of  Neptune  by  Le  Verrier — his  reasons 
for  supposing  its  existence — Mode  of  investigation — Elements 
determined  previous  to  observation — Discovery  with  the  tele- 
scope— Similar  results  obtained  by  Adams — Respective  merits 
of  these  astronomers — How  their  results  have  been  modified  by 
Walker  and  Pierce — This  planet  seen  by  Lalande  in  1795,  and 
recorded  as  a  fixed  star — Traced  back  to  this  place  by  Walker. 

Great  distance  of  Neptune  from  the  sun — Long  period  of  rev- 
olution— Magnitude,  density,  and  orbit  eccentricity. 

Appearances  of  the  different  planets  to  a  spectator  on  the  sun, 
in  regard  to  size,  light,  velocity,  &c. 


10  OUTLINES    OP    LECTURES 

LECTURE  IX. 

PLANETARY    LAWS. 

• 

Distinction  between  laws  and  individual  facts — No  laws  in  as- 
tronomy known  to  the  ancients — the  first  those  discovered  by 
Kepler — Our  knowledge  advanced  rapidly  by  laws,  slowly  by 
individual  facts.  Laws  reveal  systems  and  comprehensive  de- 
signs. 

1.  KEPLER'S  LAWS. — History  of  their  discovery — Known  to 
Kepler  as  matters  of  fact,  and  not  of  demonstration — First  dem- 
onstrated by  Newton. 

(1)  Figure  of  the  orbits — how  these  differ  from  the  eccentric 
orbits  of  the  ancient  astronomers. 

(2)  Equality  of  Areas  described  by  the  radius-vector — Great 
utility  of  this  law  in  calculating  elements. 

(3)  Distances  and  periodic  times. — Value  of  this  discovery — 
Exultation   of  Kepler  on   finding  it  —  Reduces  several   of  the 
most   important   particulars  of  the  planets  to  exact  numerical 
relations. 

2.  LAW  OF  UNIVERSAL  GRAVITATION. 

3.  NUMERICAL  RELATIONS  OP   THE  PLANETS. — Derived   from 
Kepler's  third  law  and  the  doctrine  of  universal  gravitation — 
Velocities,  distances,  periodic  times,  and  gravitation  towards  the 
sun,  all  adjusted  to  one  another,  like  the  hour,  minute,  and  second 
hands  of  a  clock — Relation  between  mass  and  velocity — Velocity, 
distance,  time,  and  force  of  gravity,  compose  a  series  in  geomet- 
rical progression  of  which  the  first  term  is  the  ratio — How  derived 
from  Kepler's  third  law  and  the  law  of  gravitation.     Any  one  of 
these  terms  being  given,  the  rest  may  be  found — illustrations — 
Method  of  finding  the  masses  and  densities  of  the  planets  ex- 
plained and  illustrated. 


LECTURE  X. 

PLANETARY    LAWS. 

1.  GREAT  MECHANICAL  LAWS  which  belong  to  every  system  of 
bodies,  connected  together  by  mutual  relations,  but  perfectly  veri- 
fied only  in  the  heavenly  bodies. 

(1)  Conservation  of  areas. 

(2)  Conservation  of  the  center  of  gravity. 

(3)  Principle  of  Least  Action. 


ON    ASTRONOMY. 


11 


2.  MOTION  IN  AN  ORBIT  UNDER  Two  FORCES. 

Experiment  with  a  suspended  ball — Effect  of  gravity  alone—- 
of the  impulsive  force  atone — Gravity  accounts  only  for  the  con- 
stant deviation  from  a  straight  line — A  primitive  impulse  indispen- 
sable— No  cause  for  it  MOW  existing — Must  have  been  applied  at 
some  previous  period  of  its  existence,  and  then  withdrawn — Argu- 
ment for  a  First  Cause — Evasion  of  this  argument — how  obviated. 

Why  a  planet  or  a  comet  turns  about  at  its  aphelion,  and  what 
keeps  it  off  from  the  sun  at  its  perihelion. 

3.  COMPREHENSIVE  VIEW  OF  THE  SOLAR  SYSTEM. 

(1)  Of  the  bodies  composing  it. 

(2)  Of  their  relations  to  one  another. 

Resemblance  of  the  subordinate  systems  to  the  whole — Uni- 
formity of  plan  throughout — General  motions  from  west  to  east — 
Exception  to  this  uniformity  in  the  satellites  of  Uranus. 

Constancy  of  the  law  of  gravitation. 

4.  STABILITY  OF  THE  SOLAR  SYSTEM. 

"  Problem  of  the  three  bodies" — Causes  of  disturbance  existing 
in  the  system — Stability  of  the  system  first  determined  by  La 
Grange,  and  confirmed  by  La  Place — Its  mathematical  expression 
[The  mathematical  formula  which  expresses  the  effect  of  all  the 
perturbations,  is  contained  in  the  sines  and  cosines  of  arcs,  which 
necessarily  vary  between  zero  and  radius] — the  perturbations 
oscillate  about  a  mean  value — Grand  axes  always  constant — also 
the  periodic  times. 

5.  PROOFS  OF  THE  COPERNICAN  SYSTEM. 
Doctrine  stated. 

1.  That  the  earth  revolves  on  its  own  axis. — More  simple — 
Analogy — Spheroidal  figure — Diminished  weight  at  the  equator 
— Bodies  fall  eastward  of  their  base. 

2.  That  the  planets,  including  the  earth,  revolve  about  the  sun. 
— Phases   of  Mercury   and  Venus — Greater   symmetry   of  the 
movements  and  orbits  of  the  superior  planets  when  referred  to 
the  sun  as  a  center. 

3.  That  the  motion  of  the  earth  around  the  sun  is  especially 
indicated — by  analogy — by  Kepler's  law — by  the  retrograde  mo- 
tions of  the  superior  planets — by  the  phenomena  of  meteors. 


LECTURE  XL 

HABITABILITY    OF    THE    PLANETS. 

Circumstances  unfavorable  to  the  doctrine — Extremes  of  neat 
and  cold — of  light — of  gravity — Want  of  atmospheres — Such 


12  OUTLINES    OF    LECTURES 

beings  as  inhabit  this  earth  could  not  inhabit  the  planets,  but  still 
some  reasons  for  thinking  that  they  have  their  appropriate  inhab- 
itants. 

1 .  Reasons  for  believing  that  the  planets  were  designed  for  the 
same  purposes  as  the  earth. 

(1)  From  the  uniformity  of  plan  manifested  throughout  nature. 

(2)  From  the  fact  that  the  earth  is  a  member  of  a  system,  and 
not  the  leading  member. 

(3)  From  the  consideration  that  all  things  have  their  use — illus- 
trations of  this  point — Uses  of  the  planets  to  our  earth  very  slight. 

In  order  to  learn  the  purpose  for  which  the  planets  were  made, 
we  inquire, 

2.  For  what  purposes  the  earth  was  made. 

(1)  Animal  life. — Its  multiplication  in  every  part  of  the  earth — 
Great  range  from  the  whale  and  the  elephant  to  the  infusoria- 
Multiplication  of  life  even  in  the  polar  seas. 

(2)  Not  animal  life  in  general,  but  Man  the  great  purpose  for 
which  the  world  was  made.     All  things  for  his  use. 

First,  the  powers  of  nature. 
Secondly,  the  productions  of  nature. 
Thirdly,  the  beauty  and  sublimity  of  nature. 
Fourthly,    the   power   of  exalting  nature — of  developing  its 
powers — of  compounding  and  forming  new  creations. 

(3)  Argument  from  analogy  applied  to  the  planets  to  prove 
that  life  is   their  general,  and  intelligent  beings  their  special 
purpose. 

(4)  Direct  evidences  that  the  planets  are  inhabited. 


LECTURE  XII. 

COMETS. 

1.  HISTORICAL  NOTICES. — Pythagoras  —  Aristotle — Tycho — 
Newton — Circumstances  that  render  them  intrinsically  interesting 
— History  of  Halley's  Comet. 

2.  LAWS  OF  THEIR  MOTIONS. — In  what  respects  they  are  alike, 
and  in  what  unlike  those  of  the  planets. 

3.  DETERMINATION  OF  THE  ORBIT  OF  A  COMET. — Nature  of  the 
difficulty — Method  of  finding  the  elements — These  may  serve  to 
identify  the  comet  with  one  that  has  appeared  before,  but  do  not 
give  the  periodic  time — How  the  time  is  ascertained — Exemplified 
in  Halley's  comet.     The  approximate  time  being  found,  to  find 
the  exact  time — Number  and  sources  of  the  perturbations — Labo- 
rious nature  of  the  calculations — Exemplified  in  the  returns  of 
Halley's  comet  in  1759  by  Lalande,  and  in  1835  by  Pontecoula^ 


ON    ASTRONOMY.  13 

How  far  the  predictions  of  astronomers,  on  the  last  return,  were 
fulfilled. 

4.  PHYSICAL  NATURE  OF  COMETS. — Constitution  of  the  three 
several  parts — Extent  and  tenuity  of  the  train — Exceedingly  small 
mass — Whether  so  rare  a  body  would  obey  the  laws  of  gravita- 
tion and  projection — Whether  the  matter  is  self-luminous — Hy- 
potheses to  account  for  the  formation  of  the  tail — by  the  impulse 
of  the  sun's  rays — by  the  action  of  heat — Difficulties  attending  all 
hypotheses  yet  proposed. 

5.  DANGERS. 

(1)  Dangers  in  various  parts  of  nature,  as  from  extremes  of 
heat  and  cold — from  the  perturbations  of  the  moon  and  planets — 
Guards  set  to  prevent  mischief — Whether  any  guards  can  be  de- 
tected in  the  case  of  comets.     Two  especially,  in  the  swiftness 
of  their  motions,  and  the  smallness  of  their  masses — also  from  the 
fact  that  they  are  subject  to  laws. 

(2)  Consequences  were  they  to  strike  the  earth — Threatening 
circumstances  attending  the  great  comet  of  1843. 

(3)  Question  of  a  resisting  medium,  and  its  final  effect  upon 
the  motions  of  comets. 


LECTURE  XIII. 

FIXED    STARS. 

Solar  system  the  chief  object  of  attention  to  astronomers  before 
Sir  William  Herschel — His  forty-feet  telescope — Rosse's  fifty-feet 
reflector — Great  refractors  recently  constructed,  as  the  Pulkova, 
the  Cincinnati,  the  Cambridge  instruments,  respectively — Com- 
parative advantages  and  disadvantages  of  reflectors  and  refractors 
— peculiar  advantages  of  large  telescopes. 

Fixed  stars. — Why  so  called — Immutability  of  the  constella- 
tions, contrasted  with  the  mutability  of  terrestrial  nature. 

Twinkling  of  the  stars  explained. 

Catalogues  of  the  stars — by  Hipparchus,  Tycho,  Flamsteed, 
Lalande — Astronomical  Society — Bessel's  zones — Berlin  charts. 

DISTANCES. — Eight  of  the  greatest  discoveries  in  astronomy  : 
1.  Kepler's  Laws.  2.  Law  of  Universal  Gravitation.  3. 
Weighing  the  Sun  and  Planets.  4.  Demonstrating  the  Stability 
of  the  Solar  System.  5.  Measuring  the  Velocity  of  Light.  6. 
Exact  Determination  of  the  Periodic  Time  of  a  Comet,  as  Hal- 
ley's.  7.  Discovery  of  the  Planet  Neptune.  8.  Measuring  the 
Distance  of  a  Fixed  Star. — Proofs  of  the  immense  distance  of 


14  OUTLINES    OP    LECTURES 

the  stars,  from  the  want  of  parallax — from  the  effect  of  the  largest 
telescopes. 

Researches  for  an  annual  parallax  by  Flamsteed,  Bradley, 
Brinkley,  Bessel — Observations  of  Bessel  on  61  Cygni — Amount 
of  its  parallax — Whether  it  may  not  be  fallacious — Illustrations 
of  the  distance  of  61  Cygni  by  the  time  occupied  by  light — bv  a 
railway-car. 

Supposed  parallax  of  Alpha  Centauri — of  Sirius. 


LECTURE  XIV. 

SYSTEMS  OF  STARS NATURE  OF  THE  STARS FINAL  PURPOSE. 

1.  SYSTEMS  OF  STARS. 

(1)  Double,  triple,  and  multiple  stars. — Great  number  of  such 
groups — Sometimes  merely  optically  double — Revolutions  of  the 
binary  stars — Proof  which  they  afford  of  the  extent  of  the  law 
of  gravitation. 

(2)  Clusters. — Pleiades — Beehive — in  the  head  of  Orion — in 
the  sword-handle  of  Perseus. 

(3)  Nebula. — Description — Various  forms — Extent — Resolva- 
ble and  irresolvable — Whether  this  distinction  is  absolute  or  rel- 
ative— Bearing  of  recent  discoveries  of  the  great  telescopes  on 
this  question — Researches  of  Mason  and  Smith  on  nebulae — Ob- 
servations of  Sir  John  Herschel  in  the  southern  hemisphere. 

(4)  Galaxy. — Its  figure — Constitution  according  to  Sir  William 
Herschel — The  relation  of  our  solar  system  to  it — How  our  posi- 
tion in  it  is  determined. 

2.  NATURE  OF  THE  STARS. 

(1)  Material  bodies — obey  the  law  of  attraction. 

(2)  Larger  than  our  earth — Sum  of  the  masses  of  both  the  stars 
in  61  Cygni  half  that  of  the  sun — Orbit  twice  that  of  Uranus — 
Period  540  years — Sirius  equal  to  two  suns,  or,  according  to  Wol- 
laston's  experiments  upon  its  intrinsic  light,  to  fourteen  suns. 

(3)  Light  of  all  has  the  same  velocity — Velocity  uniform — 
Constant  of  aberration  the  same  in  all. 

(4)  Light  of  the  stars  not  polarized,  therefore  direct,  and  not 
reflected. 

3.  FINAL  PURPOSE  of  the  stars. 

Not  for  ornament — not  to  give  light  by  night — not  for  naviga- 
tion— Are  suns  of  other  systems — Uniformity  of  plan  in  the  works 
of  creation — Purpose  of  our  sun  for  light,  heat,  and  attraction  to 
planetary  worlds — Life  the  great  object  of  these — All  for  intelli- 
gent, immortal  beings. 


ON   ASTRONOMY.  15 

LECTURE  XV. 

MECHANISM  OP  THE  HEAVENS. 

L  Propriety  of  reasoning  from  the  uniformity  of  plan  in  nature 
— Deducing  what  we  cannot  see  from  what  we  do  see. 

2.  What  we  actually  see. 

(1)  Subordinate  systems,  as  Jupiter  and  his  satellites. 

(2)  The  solar  system. 

(3)  The  binary  stars. 

(4)  Higher  systems  in  groups,  clusters,  and  nebulae. 

3.  Probable   prevalence  of  the  law  of  universal  gravitation 
through  the  universe. — Consequences  of  such  a  law — Analogy 
leads  us  to  expect  revolutions — Mere  attraction  without  it  tends 
to  ruin — Proper  motions  detected  among  the  stars — These  mo- 
tions of  revolution. 

4.  Grand  machine  of  the  universe. — Probability  that  it  is  con- 
structed after  the  model  of  the  solar  system — Its  actual  organi- 
zation. 


LECTURE  XVI. 

5UN NEBULAR  HYPOTHESIS CONCLUDING 

REFLECTIONS. 

CENTER  OF  THE  UNIVERSE  (?) — Attempts  of  Maedlerto  find  the 
center  about  which  the  stars  of  our  firmament  revolve — Fixed  in 
Alcyone,  the  principal  star  of  the  Pleiades — Reasons  for  selecting 
this  point — Whether  any  thing  more  is  necessary  than  a  common 
center  of  gravity — Movement  of  the  solar  system,  and  of  various 
other  stellar  systems  in  respect  to  this  point. 

NEBULAR  HYPOTHESIS. 

1.  Herschel's  views  of  the  progressive  state  of  the  nebulae — 
Whether  changes  now  take  place  within  short  periods — Mason's 
researches  on  this  subject. 

2.  La  Place's  hypothesis  of  the  formation  of  the  solar  system 
— Principal  object  to  account  for  the  revolutions  all  in  the  same 
direction— Principles  of  it  stated — Supposed  illustration  by  ex- 
periment— Phenomena  explained  by  the  hypothesis — Comets  not 
included — Objections  considered — whether  the  doctrine  is  neces- 
sarily atheistical — whether  such  a  mode  of  formation  is  possible — 
whether  probable — Backward  motion  of  the  satellites  of  Uranus 
— Bearing  on  this  hypothesis  of  late  telescopic  discoveries  among 
ths  nebulas. 


16  OUTLINES  OP  LECTURES  ON  ASTRONOMY. 

CONCLUDING  REFLECTIONS. 

1.  Relations  of  Astronomy  to  Natural  Theology. 

2.  Place  which  our  world  and  which  man  holds  in  the  universe 
— Supposition  that  both  are  too  insignificant  to  be  objects  of  high 
and  peculiar  interest  to  the  Creator,  considered 

3.  View  which  the  Creator  took  of  his  own  works  when  be 
pronounced  them  "  very  good." 


1 

PH 


PLATE  II. 
NEBULA  AND   DOUBLE  STARS, 


1.  Castor.     2.  7  Leonis.  3.  39  Drac.     4.  X  Oph.  5.  11  Monoc.  G  £  Cancri. 


Revolutions  of  y  Virginis. 


n 
1837.   1838.   1839.   1840.   1845.   1850.  1860.   Orbit. 


RECOMMENDATIONS  OF  "COFFIN'S  CONIC  SECTIONS/' 


FROM  PROF.  MASON,  BETHANY  COLLEGE,  VA. 

I  have  examined  "  Coffin's  Conic  Sections  and  Analytical  Geometry,"  and  hesitate  not 
to  say,  that  its  size,  plan  and  general  arrangement,  ought  to  procure  for  it  a  place  in  the 
hands  of  every  teacher  who  wishes  to  give  a  liberal  knowledge  of  Mathematics,  without 
abridging  the  other  branches  of  a  good  education.  I  am  happy  to  say  that  your  book  is 
better  adapted  to  my  present  wants  than  any  I  have  seen.  I  have  therefore  concluded 
to  adopt  it  for  my  next  class. 

FROM  J.  S.  LEBAR,  PRINCIPAL  OF  THE  HIGH  SCHOOL,  HACKETTSTOWN,  N.  J. 
By  bringing  out  this  work  you  have  advanced  the  cause  of  science,  merited  the  thanks 

)f  the  lovers  of  learning,  and  done  a  lasting  good  to  the  public As  a  general  principle, 

f  am  opposed  to  the  shortening  of  books  in  order  to  shorten  the  course  of  study.  The  spirit 
)f  the  day  is  to  get  along  most  rapidly,  not  most  thoroughly  ;  by  bestowing  the  least  pos- 
sible labor,  both  in  physical  and  mathematical  science.  But  from  the  examination  I  have 
been  able  to  make  of  your  work,  I  find  it  contains  some  important  matter,  not  found  in 
other  works  of  a  like  character,  whilst  nothing  that  is  valuable  is  omitted.  The  compass  is 
contracted,  but  the  whole  circle  is  there.  It  is  just  the  work  we  need,  and  I  shall  adopt  it 

in  our  Academy  in  preference  to  other  treatises  upon  these  subjects Certainly  by 

your  combining  the  geometrical  and  analytical  methods,  you  present  the  subject  more 

clearly Your  demonstrations  are  admirable  ;  they  are  so  concise,  yet  so  simple,  that 

no  student  can  pass  over  them  without  fully  comprehending  them. 

FROM  PROF.  SUDLER,  DICKINSON  COLLEGE,  PENN. 

I  have  received  and  read  with  great  pleasure  your  "  Elements  of  Conic  Sections  and 
Analytical  Geometry,"  and  cannot  hesitate  to  commend  it  as  an  excellent  system.  It 
embraces  in  a  parallel  view,  the  Geometrical  Theory  and  the  Analytical  investigation  of 
the  properties  of  the  Conic  Sections — the  method  which  I  have  always  pursued  in  the 
instruction  in  my  department,  but  have  felt  the  want  of  a  proper  text-book  on  the  subject 
This  want  has  been  met  by  your  Treatise,  and  I  have  adopted  it  as  a  text-book  with 
pleasure. 


FROM  PROF.  FOUCHE,  ST.  JOHN'S  COLLEGE,  N.  Y. 

Your  work  was  examined,  I  do  not  say  by  me,  but  by  judges  more  competent  than  I 
am.  We  are  unanimous  in  saying  that  your  demonstrations,  with  respect  to  perspicuity, 
are  to  be  highly  praised.  You  are  clear,  concise,  and,  whilst  you  avoid  obscurity,  you 
find  the  means  of  being  both  exact  and  brief.  This  is  no  small  merit,  and  you  are  enti- 
tled to  the  gratitude  of  students,  to  whom  you  spare  useless  difficulties  and  exertions. 


FROM  PROF.  LOOMIS,  COLLEGE  OF  NEW  JERSEY,  PRINCETON,  N.  J. 

Your  treatise  on  "  Analytical  Geometry"  Appears  to  me  to  possess  advantages  over 
any  other  treatise  which  I  have  examined.  vOne  of  these  advantages  I  consider  to  be, 
the  division  into  Propositions  distinctly  enunciated,  so  as  to  keep  constantly  before  the 
mind  of  the  student  the  object  of  his  investigation.  Another  advantage  consists  in  the 
introduction  of  numerical  examples,  which  afford  a  useful  exercise  to  the  student,  and 
present  the  best  test  of  the  clearness  of  his  conceptions. 

FROM  J.  S.  ALVERSON,  PRINCIPAL  OF  THE  GENESEE  WESLEYAN  SEMINARY,  N.  Y. 

I  have  received  and  examined  a  copy  of  your  recent  work  on  "  Conic  Sections."  .... 
The  manner  in  which  you  have  introduced  both  the  geometrical  and  analytical  demon- 
strations, ought  to  secure  success  to  your  work.  You  present  what  I  think  is  needed  by 
the  American  student.  In  the  selection  of  problems  and  in  the  elucidation  of  parts  that 
often  prove  unreasonably  obscure  to  beginners,  you  have  been  very  happy I  recom- 
mend the  work  with  confidence  to  the  attention  of  Instructors  and  Students. 


RECOMMENDATIONS. 

FROM  PROF.  STEVENS,  OAKLAND  COLLEGE,  Miss. 

I  sent  to  New  Orleans  for  a  copy  of  your  treatise  on  the  "  Conic  Sections  and  Analyt- 
ical Geometry,"  which  I  have  received  and  have  examined  with  some  care  and  much 
satisfaction.  I  think  after  the  perusal  I  have  given  it,  I  may  safeJy  pronounce  your  trea- 
tise eminently  suited  for  use  in  colleges ;  it  is  sufficiently  elaborate,  the  arrangement  i& 
good,  and  there  is  a  plainness  and  simplicity  about  it  which  I  admire  very  much. 


MESSRS.  COLLINS  &  BROTHER. — Gent. :  Illness  has  prevented  me  from  examining 
Prof.  Coffin's  "  Treatise  on  Conic  Sections"  till  very  recently.  I  am  greatly  pleased  with 
the  work,  and  think  that  the  author  has  succeeded  in  developing  the  most  interesting  and 
important  properties  of  these  curves  with  unusual  simplicity  and  brevity  in  the  demon- 
strations. To  this,  the  common  property  of  the  curves,  assumed  as  their  fundamental 
characteristic,  is,  by  the  skill  and  tact  of  the  author,  made  to  contribute  very  much. 
This  property,  while  it  unites  the  three  curves  in  a  common  bond,  and  gives  them  a 
common  source,  is  that  on  which  the  more  useful  properties  seem  most  immediately  to 
depend,  and  from  which  they  are  most  readily  and  naturally  deduced.  In  the  Geometri- 
cal portion  there  is  a  very  just  medium  preserved,  in  the  extent  to  which  the  discussion 
of  the  curves  is  carried,  the  investigations  being  limited  to  the  most  essential  and  practi- 
cally useful  properties,  and  requiring  no  useless  expenditure  of  time  on  points  merely  spec- 
ulative or  curious.  The  second  part,  devoted  to  "  Analytical  Geometry,"  is  a  very  im- 
portant addition  to  the  work,  and  one,  I  think,  without  which  the  student  will  be  very 
illy  prepared  to  make  his  knowledge  of  these  curves,  readily  and  extensively  available  in 
their  application  to  astronomical  investigations.  The  work  I  regard  as.  a  very  decided 
improvement  on  the  old  systems  and  treatises,  and  think  in  its  preparation  and  publication 
a  very  valuable  contribution  has  been  made  to  educational  instrumentalities. 

Very  respectfully  yours, 

WESLEYAN  UNIVERSITY,  April  20,  1849.  AUG.  W.  SMITH. 


FROM  THE  METHODIST  QUARTERLY  REVIEW. 

"  Elements  of  the  Conic  Sections  and  Analytical  Geometry,  by  JAMES  H.  COFFIN,  A.  M. 

Professor  of  Mathematics,  $-c.,  in  Lafayette  College" 

In  this  treatise  the  doctrine  of  the  Conic  Sections  is  taught  first  geometrically  ;  and  in 
a  second  part,  the  student  is  taught  how  to  represent  lines,  curves,  and  surfaces  analyti- 
cally, and  to  solve  problems  relating  to  them.  We  have  examined  the  work  with  care, 
and  testify  to  the  skill,  tact,  and  neatness  of  its  expositions.  Most  books  of  Analytical 
Geometry  are  blind  to  scholars  ;  most  of  them  never  learn,  unless  they  have  a  teacher 
unusually  skilful  and  diligent,  how  to  interpret  algebraical  expressions,  or  how  to  make 
practical  use  of  equations.  It  is  precisely  for  its  clearness,  its  practical  character,  and  its 
adaptation  to  the  work  of  the  recitation  room,  that  we  heartily  commend  this  volume. 


FROM  J.  F.  JENKINS,  PRINCIPAL  OF  THE  NORTH  SALEM  ACADEMY. 

I  have  recently  had  an  opportunity  of  examining  your  treatise  on  "  Conic  Sections  and 
Analytical  Geometry,"  and  am  happy  to  state  that,  in  my  opinion,  it  is  a  work  of  much 
merit.  The  selection  and  arrangement  of  the  Propositions  appear  to  me  judicious ;  and 
the  definition  which  you  adopt,  at  the  commencement,  has  certainly  the  advantage  of 
supplying  more  simple  and  obvious  demonstrations,  in  place  of  those  whose  prolixity  and 
abstruseness  have  hitherto  rendered  this  very  important  portion  of  Mathematics  so  for- 
midable to  learners.  The  first  part  appears  to  contain  all  the  properties  of  the  Sections 
which  are  required  in  the  branches  of  Mathematical  Philosophy  usually  embraced  in  our 
College  courses ;  and  the  second  part  furnishes  a  valuable  introduction  to  Analytical  Ge- 
ometry, rendered  more  valuable  by  its  immediate  connection  with  what  precedes,  and  by 
the  practical  nature  of  the  problems  proposed  for  investigation  by  analysis.  I  intend  in- 
troducing it  as  a  text-book  in  this  Institution. 


FROM  JAMES  T.  DORAN,  TEACHER,  MANALAPAN,  N.  J 
I  think  it  decidedly  the  best  work  in  print  upon  the  subject. 


ROBERT  B,  COLLINS, 

BOOKSELLER  AND  STATIONER,  254  PEARL-STREET,  N.  Y. 

Invites  the  attention  of  those  interested  to  the  following  valuable  books,  which,  it 
is  bt'lievi'd,  will  be  found  on  examination  unsurpassed  as  text  books. 

MATHEMATICS.—  ADAMS'S  ARITHMETICAL  SERIES. 

I.  Primary  Arithmetic.—  An   introduction   to  Adams's  New  Arithmetic,  revised  edition. 

II.  Adam*'*  New  Arithmetic,  revised  edition;  being  a  revision  of  Adams's  New  Arith- 
metic, lirst  published  in  1«^7.    liino.  halt  bound. 

A  Key  to  the  above  is  published  separately. 

III.  Mensuration,  Mechanical  Towers,  and  Machinery,  a  sequel  to  the  Arithmetic. 

IV.  Bookkeeping  by  Single  Entry,  accompanied  with  blank  books  for  the  use  of  learners. 

Abbott's  Arithmetic. 

The  II  on  nt  Vernon  Arithmetic.—  Part  I.,  Elementary.    By  Jacob  and  Charles  E.  Abbott 
The  Mount  Vernon  Arithmetic.—  Part  II.,  Vulgar  and  Decimal  Fractions.     By  Jacob  Ab- 
bott.    12mo.,  half  bound. 

McCnrdy's  Geometry. 

First  Lessons  in  Geometry.     By  D.  McCurdy. 

Charts  to  accompany  the  "First  Lessons,"  on  rollers,  size  34  by  48  inches. 
Euclid's  Elements,  or  Second  Lessons  in  Geometry.     By  D.  McCurdy.    12mo.  half  bound. 

Preston's  Bookkeeping. 
District  School  Bookkeeping.    Quarto,  stitched.    An  excellent  work  for  beginners  ;  printed 

on  handsome  demy  paper  for  practice. 
Single  Entry  Bookkeeping.    8vo.    cloth  sides. 
Bookkeeping,  by  single  and  double  entry. 

These  three  excellent  works  are  by  Lyman  Preston,  the  author  of  the  Interest  Tables,  &c. 

Coffin's  Conic  Sections. 

Elements  of  Conic  Sections  and  Analytical  Geometry.     By  James  H.  Coffin,  Prof.  Mathe- 
matics, Lafayette  College,  Pa. 

Day's  Mathematics. 

I  Course  of  Mathematics,  containing  the  principles  of  Plane  Trigonometry,  Mensuration, 
Navigation,  and  Surveying.    By  Jeremiah  Day,  D.  D.  LL.  D.,  President  of  Yale  Col.  8vo.  sheep. 

1  1ILOSOPHY  AND  ASTRONOMY. 

<)  I  nisi  «•<:'*  Rudiments  of  Natural  Philosophy.    18mo. 
Rudiments  of  Astronomy.    18mo. 


These  two  works  are  intended  to  be  used  in  district  and  common  schools,  where  merely  the 
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Olmstcd's  College  Philosophy,    hvo.,  sheep. 

These  works  were  prepared  by  Prof.  Olmsted  for  the  use  of  bis  classes  in  Yale  College,  and 
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MISCELLANEOUS. 

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Gnbriel,  a  story  of  Wichnor  Wood.     By  Mary  Howitt.    The  best  taleof  this  popular  authoress. 
Our  Cousins  in  Ohio,  a  picture  of  domestic  life  in  the  West.    By  Mary  Howitt.  18mo.,  cloth  gilt. 
The  Imitation  of  Christ.      Hy  T.  a  Kempis.    Payne's  translation,  with   essay  by  Dr.  Chal- 

mers.   The  cheapest  edition  of  this  valuable  work  ever  issued  from  the  press. 
Abercrombie's  Intellectual  Powers.—  Inquiries  concerning  the  Intellectual  Powers,  and 

In  ve-tigatum  of  Truth.    By  John  Abercrombie.    12mo.,  cloth. 
Abercrombie's  Moral  Philosophy. 

These  two  works  of  Abercroinbie'si  are  made  much  more  valuable  by  the  additions  and  ques- 

tions to  I  his  edition  by  Jacob  Abbott. 
D>moud's  Essays  on  the  Principles  of  Morality,  and  the  private  and  political  rights  and  obliga- 

tions of  Mankind.    This  is  regarded  as  the  best  work  on  Ethics  in  the  language.    12rao. 


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